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# R.D. Sharma solutions for Class 12 Mathematics chapter 27 - Direction Cosines and Direction Ratios

## Chapter 27 - Direction Cosines and Direction Ratios

#### Pages 22 - 23

Q 1 | Page 22

If a line makes angles of 90°, 60° and 30° with the positive direction of xy, and z-axis respectively, find its direction cosines

Q 2 | Page 23

If a line has direction ratios 2, −1, −2, determine its direction cosines.

Q 3 | Page 23

Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3) .

Q 4 | Page 23

Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.

Q 5 | Page 23

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, −4), (−1, 1, 2) and (−5, −5, −2).

Q 6 | Page 23

Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.

Q 7 | Page 23

Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.

Q 8 | Page 23

Find the acute angle between the lines whose direction ratios are proportional to 2 : 3 : 6 and 1 : 2 : 2.

Q 9 | Page 23

Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.

Q 10 | Page 23

Show that the line through points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (−1, −2, 1) and (1, 2, 5).

Q 11 | Page 23

Show that the line through the points (1, −1, 2) and (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Q 12 | Page 23

Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, −1) and (4, 3, −1).

Q 13 | Page 23

Find the angle between the lines whose direction ratios are proportional to abc and b − cc − aa− b.

Q 14 | Page 23

If the coordinates of the points ABCD are (1, 2, 3), (4, 5, 7), (−4, 3, −6) and (2, 9, 2), then find the angle between AB and CD.

Q 15 | Page 23

Find the direction cosines of the lines, connected by the relations: l + m +n = 0 and 2lm + 2ln − mn= 0.

Q 16.1 | Page 23

Find the angle between the lines whose direction cosines are given by the equations
(i) m + n = 0 and l2 + m2 − n2 = 0

Q 16.2 | Page 23

Find the angle between the lines whose direction cosines are given by the equations

2l − m + 2n = 0 and mn + nl + lm = 0

Q 16.3 | Page 23

Find the angle between the lines whose direction cosines are given by the equations

l + 2m + 3n = 0 and 3lm − 4ln + mn = 0

Q 16.4 | Page 23

Find the angle between the lines whose direction cosines are given by the equations

2l + 2m − n = 0, mn + ln + lm = 0

#### Pages 24 - 25

Q 1 | Page 24

Define direction cosines of a directed line.

Q 2 | Page 24

What are the direction cosines of X-axis?

Q 3 | Page 24

What are the direction cosines of Y-axis?

Q 4 | Page 24

What are the direction cosines of Z-axis?

Q 5 | Page 24

Write the distances of the point (7, −2, 3) from XYYZ and XZ-planes.

Q 6 | Page 24

Write the distance of the point (3, −5, 12) from X-axis?

Q 7 | Page 24

Write the ratio in which YZ-plane divides the segment joining P (−2, 5, 9) and Q (3, −2, 4).

Q 8 | Page 24

A line makes an angle of 60° with each of X-axis and Y-axis. Find the acute angle made by the line with Z-axis.

Q 9 | Page 25

If a line makes angles α, β and γ with the coordinate axes, find the value of cos 2α + cos 2β + cos 2γ.

Q 10 | Page 25

Write the ratio in which the line segment joining (abc) and (−a, −c, −b) is divided by the xy-plane.

Q 11 | Page 25

Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.

Q 12 | Page 25

Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.

Q 13 | Page 25

Write the distance of the point P (xyz) from XOY plane.

Q 14 | Page 25

Write the coordinates of the projection of point P (xyz) on XOZ-plane.

Q 15 | Page 25

Write the coordinates of the projection of the point P (2, −3, 5) on Y-axis.

Q 16 | Page 25

Find the distance of the point (2, 3, 4) from the x-axis.

Q 17 | Page 25

If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?

Q 18 | Page 25

Write direction cosines of a line parallel to z-axis.

Q 19 | Page 25

If a unit vector  vec a makes an angle $\frac{\pi}{3} \text{ with } \hat{i} , \frac{\pi}{4} \text{ with } \hat{j}$ and an acute angle θ with $\hat{ k}$ ,then find the value of θ.

Q 20 | Page 25

Answer each of the following questions in one word or one sentence or as per exact requirement of the question:
Write the distance of a point P(abc) from x-axis.

Q 21 | Page 25

If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.

#### Pages 25 - 26

Q 1 | Page 25

For every point P (xyz) on the xy-plane,

x = 0

y = 0

z = 0

x = y = z = 0

Q 2 | Page 25

For every point P (xyz) on the x-axis (except the origin),

x = 0, y = 0, z ≠ 0

x = 0, z = 0, y ≠ 0

y = 0, z = 0, x ≠ 0

x = y = z = 0

Q 3 | Page 25

A rectangular parallelopiped is formed by planes drawn through the points (5, 7, 9) and (2, 3, 7) parallel to the coordinate planes. The length of an edge of this rectangular parallelopiped is

2

3

4

all of these

Q 4 | Page 25

A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7), parallel to the coordinate planes. The length of a diagonal of the parallelopiped is

7

sqrt(38)

sqrt(155)

none of these

Q 5 | Page 25

The xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)

internally in the ratio 2 : 3

externally in the ratio 2 : 3

internally in the ratio 3 : 2

externally in the ratio 3 : 2

Q 6 | Page 25

If the x-coordinate of a point P on the join of Q (2, 2, 1) and R (5, 1, −2) is 4, then its z-coordinate is

2

1

-1

-2

Q 7 | Page 25

The distance of the point P (abc) from the x-axis is

$\sqrt{b^2 + c^2}$

$\sqrt{a^2 + c^2}$

$\sqrt{a^2 + b^2}$

none of these

Q 8 | Page 26

Ratio in which the xy-plane divides the join of (1, 2, 3) and (4, 2, 1) is

3 : 1 internally

3 : 1 externally

1 : 2 internally

2 : 1 externally

Q 9 | Page 26

If P (3, 2, −4), Q (5, 4, −6) and R (9, 8, −10) are collinear, then R divides PQ in the ratio

3 : 2 externally

3 : 2 internally

2 : 1 internally

2 : 1 externally

Q 11 | Page 26

If O is the origin, OP = 3 with direction ratios proportional to −1, 2, −2 then the coordinates of P are

(−1, 2, −2)

(1, 2, 2)

(−1/9, 2/9, −2/9)

(3, 6, −9)

Q 12 | Page 26

The angle between the two diagonals of a cube is

(a) 30°

(b) 45°

(c) $\cos^{- 1} \left( \frac{1}{\sqrt{3}} \right)$

(d) $\cos^{- 1} \left( \frac{1}{3} \right)$

Q 13 | Page 26

If a line makes angles α, β, γ, δ with four diagonals of a cube, then cos2 α + cos2 β + cos2γ + cos2 δ is equal to

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{4}{3}$

$\frac{8}{3}$

#### Pages 9 - 10

Q 1 | Page 9

Find the vector and cartesian equations of the line through the point (5, 2, −4) and which is parallel to the vector  $3 \hat{i} + 2 \hat{j} - 8 \hat{k} .$

Q 2 | Page 9

Find the vector equation of the line passing through the points (−1, 0, 2) and (3, 4, 6).

Q 3 | Page 9

Find the vector equation of a line which is parallel to the vector $2 \hat{i} - \hat{j} + 3 \hat{k}$  and which passes through the point (5, −2, 4). Also, reduce it to cartesian form.

Q 4 | Page 9

A line passes through the point with position vector $2 \hat{i} - 3 \hat{j} + 4 \hat{k}$ and is in the direction of  $3 \hat{i} + 4 \hat{j} - 5 \hat{k} .$ Find equations of the line in vector and cartesian form.

Q 5 | Page 9

ABCD is a parallelogram. The position vectors of the points AB and C are respectively, $4 \hat{ i} + 5 \hat{j} -10 \hat{k} , 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \text{ and } - \hat{i} + 2 \hat{j} + \hat{k} .$  Find the vector equation of the line BD. Also, reduce it to cartesian form.

Q 6 | Page 9

Find in vector form as well as in cartesian form, the equation of the line passing through the points A (1, 2, −1) and B (2, 1, 1).

Q 7 | Page 9

Find the vector equation for the line which passes through the point (1, 2, 3) and parallel to the vector $\hat{i} - 2 \hat{j} + 3 \hat{k} .$  Reduce the corresponding equation in cartesian from.

Q 8 | Page 10

Find the vector equation of a line passing through (2, −1, 1) and parallel to the line whose equations are $\frac{x - 3}{2} = \frac{y + 1}{7} = \frac{z - 2}{- 3} .$

Q 9 | Page 10

The cartesian equations of a line are $\frac{x - 5}{3} = \frac{y + 4}{7} = \frac{z - 6}{2} .$  Find a vector equation for the line.

Q 10 | Page 10

Find the cartesian equation of a line passing through (1, −1, 2) and parallel to the line whose equations are  $\frac{x - 3}{1} = \frac{y - 1}{2} = \frac{z + 1}{- 2}$  Also, reduce the equation obtained in vector form.

Q 11 | Page 10

Find the direction cosines of the line  $\frac{4 - x}{2} = \frac{y}{6} = \frac{1 - z}{3} .$  Also, reduce it to vector form.

Q 12 | Page 10

The cartesian equations of a line are x = ay + bz = cy + d. Find its direction ratios and reduce it to vector form.

Q 13 | Page 10

Find the vector equation of a line passing through the point with position vector  $\hat{i} - 2 \hat{j} - 3 \hat{k}$  and parallel to the line joining the points with position vectors  $\hat{i} - \hat{j} + 4 \hat{k} \text{ and } 2 \hat{i} + \hat{j} + 2 \hat{k} .$ Also, find the cartesian equivalent of this equation.

Q 14 | Page 10

Find the points on the line $\frac{x + 2}{3} = \frac{y + 1}{2} = \frac{z - 3}{2}$  at a distance of 5 units from the point P (1, 3, 3).

Q 15 | Page 10

Show that the points whose position vectors are  $- 2 \hat{i} + 3 \hat{j} , \hat{i} + 2 \hat{j} + 3 \hat{k} \text{ and } 7 \text{ i} - \text{ k}$  are collinear.

Q 16 | Page 10

Find the cartesian and vector equations of a line which passes through the point (1, 2, 3) and is parallel to the line  $\frac{- x - 2}{1} = \frac{y + 3}{7} = \frac{2z - 6}{3} .$

Q 17 | Page 10

The cartesian equation of a line are 3x + 1 = 6y − 2 = 1 − z. Find the fixed point through which it passes, its direction ratios and also its vector equation.

Q 18 | Page 10

Find the vector equation of the line passing through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7y = 35z.

#### Pages 15 - 17

Q 1 | Page 15

Show that the three lines with direction cosines $\frac{12}{13}, \frac{- 3}{13}, \frac{- 4}{13}; \frac{4}{13}, \frac{12}{13}, \frac{3}{13}; \frac{3}{13}, \frac{- 4}{13}, \frac{12}{13}$ are mutually perpendicular.

Q 2 | Page 15

Show that the line through the points (1, −1, 2) and (3, 4, −2) is perpendicular to the through the points (0, 3, 2) and (3, 5, 6).

Q 3 | Page 16

Show that the line through the points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (−1, −2, 1) and, (1, 2, 5).

Q 4 | Page 16

Find the cartesian equation of the line which passes through the point (−2, 4, −5) and parallel to the line given by  $\frac{x + 3}{3} = \frac{y - 4}{5} = \frac{z + 8}{6} .$

Q 5 | Page 16

Show that the lines $\frac{x - 5}{7} = \frac{y + 2}{- 5} = \frac{z}{1} and \frac{x}{1} = \frac{y}{2} = \frac{z}{3}$  are perpendicular to each other.

Q 6 | Page 16

Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, −1) and (4, 3, −1).

Q 7 | Page 16

Find the equation of a line parallel to x-axis and passing through the origin.

Q 8.1 | Page 16

Find the angle between the following pairs of lines:

$\vec{r} = \left( 4 \hat{i} - \hat{j} \right) + \lambda\left( \hat{i} + 2 \hat{j} - 2 \hat{k} \right) \text{ and }\vec{r} = \hat{i} - \hat{j} + 2 \hat{k} - \mu\left( 2 \hat{i} + 4 \hat{j} - 4 \hat{k} \right)$

Q 8.2 | Page 16

Find the angle between the following pairs of lines:

$\vec{r} = \left( 3 \hat{i} + 2 \hat{j} - 4 \hat{k} \right) + \lambda\left( \hat{i} + 2 \hat{j} + 2 \hat{k} \right) \text{ and } \vec{r} = \left( 5 \hat{j} - 2 \hat{k} \right) + \mu\left( 3 \hat{i} + 2 \hat{j} + 6 \hat{k} \right)$

Q 8.3 | Page 16

Find the angle between the following pairs of lines:

$\vec{r} = \lambda\left( \hat{i} + \hat{j} + 2 \hat{k} \right) \text{ and } \vec{r} = 2 \hat{j} + \mu\left\{ \left( \sqrt{3} - 1 \right) \hat{i} - \left( \sqrt{3} + 1 \right) \hat{j} + 4 \hat{k} \right\}$

Q 9.1 | Page 16

Find the angle between the following pairs of lines:

$\frac{x + 4}{3} = \frac{y - 1}{5} = \frac{z + 3}{4} and \frac{x + 1}{1} = \frac{y - 4}{1} = \frac{z - 5}{2}$

Q 9.2 | Page 16

Find the angle between the following pairs of lines:

$\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{- 3} and \frac{x + 3}{- 1} = \frac{y - 5}{8} = \frac{z - 1}{4}$

Q 9.3 | Page 16

Find the angle between the following pairs of lines:

$\frac{5 - x}{- 2} = \frac{y + 3}{1} = \frac{1 - z}{3} and \frac{x}{3} = \frac{1 - y}{- 2} = \frac{z + 5}{- 1}$

Q 9.4 | Page 16

Find the angle between the following pairs of lines:

$\frac{x - 2}{3} = \frac{y + 3}{- 2}, z = 5 \text{ and } \frac{x + 1}{1} = \frac{2y - 3}{3} = \frac{z - 5}{2}$

Q 9.5 | Page 16

Find the angle between the following pairs of lines:

$\frac{x - 5}{1} = \frac{2y + 6}{- 2} = \frac{z - 3}{1} and \frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 6}{5}$

Q 9.6 | Page 16

Find the angle between the following pairs of lines:

$\frac{- x + 2}{- 2} = \frac{y - 1}{7} = \frac{z + 3}{- 3} and \frac{x + 2}{- 1} = \frac{2y - 8}{4} = \frac{z - 5}{4}$

Q 10.1 | Page 16

Find the angle between the pairs of lines with direction ratios proportional to
(i) 5, −12, 13 and −3, 4, 5

Q 10.2 | Page 16

Find the angle between the pairs of lines with direction ratios proportional to  2, 2, 1 and 4, 1, 8 .

Q 10.3 | Page 16

Find the angle between the pairs of lines with direction ratios proportional to  1, 2, −2 and −2, 2, 1 .

Q 10.4 | Page 16

Find the angle between the pairs of lines with direction ratios proportional to   abc and b − cc − aa − b.

Q 11 | Page 16

Find the angle between two lines, one of which has direction ratios 2, 2, 1 while the  other one is obtained by joining the points (3, 1, 4) and (7, 2, 12).

Q 12 | Page 16

Find the equation of the line passing through the point (1, 2, −4) and parallel to the line $\frac{x - 3}{4} = \frac{y - 5}{2} = \frac{z + 1}{3} .$

Q 13 | Page 16

Find the equations of the line passing through the point (−1, 2, 1) and parallel to the line  $\frac{2x - 1}{4} = \frac{3y + 5}{2} = \frac{2 - z}{3} .$

Q 14 | Page 16

Find the equation of the line passing through the point (2, −1, 3) and parallel to the line  $\vec{r} = \left( \hat{i} - 2 \hat{j} + \hat{k} \right) + \lambda\left( 2 \hat{i} + 3 \hat{j} - 5 \hat{k} \right) .$

Q 15 | Page 16

Find the equations of the line passing through the point (2, 1, 3) and perpendicular to the lines  $\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z - 3}{3} and \frac{x}{- 3} = \frac{y}{2} = \frac{z}{5}$

Q 16 | Page 17

Find the equation of the line passing through the point  $\hat{i} + \hat{j} - 3 \hat{k}$ and perpendicular to the lines  $\vec{r} = \hat{i} + \lambda\left( 2 \hat{i} + \hat{j} - 3 \hat{k} \right) \text { and } \vec{r} = \left( 2 \hat{i} + \hat{j} - \hat{ k} \right) + \mu\left( \hat{i} + \hat{j} + \hat{k} \right) .$

Q 17 | Page 17

Find the equation of the line passing through the point (1, −1, 1) and perpendicular to the lines joining the points (4, 3, 2), (1, −1, 0) and (1, 2, −1), (2, 1, 1).

Q 18 | Page 17

Determine the equations of the line passing through the point (1, 2, −4) and perpendicular to the two lines $\frac{x - 8}{8} = \frac{y + 9}{- 16} = \frac{z - 10}{7} and \frac{x - 15}{3} = \frac{y - 29}{8} = \frac{z - 5}{- 5}$

Q 19 | Page 17

Show that the lines $\frac{x - 5}{7} = \frac{y + 2}{- 5} = \frac{z}{1} \text{ and } \frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ are perpendicular to each other.

Q 20 | Page 17

Find the vector equation of the line passing through the point (2, −1, −1) which is parallel to the line 6x − 2 = 3y + 1 = 2z − 2.

Q 21 | Page 17

If the lines $\frac{x - 1}{- 3} = \frac{y - 2}{2 \lambda} = \frac{z - 3}{2} and \frac{x - 1}{3\lambda} = \frac{y - 1}{1} = \frac{z - 6}{- 5}$  are perpendicular, find the value of λ.

Q 22 | Page 17

If the coordinates of the points ABCD be (1, 2, 3), (4, 5, 7), (−4, 3, −6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD

Q 23 | Page 17

Find the value of λ so that the following lines are perpendicular to each other. $\frac{x - 5}{5\lambda + 2} = \frac{2 - y}{5} = \frac{1 - z}{- 1}, \frac{x}{1} = \frac{2y + 1}{4\lambda} = \frac{1 - z}{- 3}$

Q 24 | Page 17

Find the direction cosines of the line

$\frac{x + 2}{2} = \frac{2y - 7}{6} = \frac{5 - z}{6}$  Also, find the vector equation of the line through the point A(−1, 2, 3) and parallel to the given line.

#### Pages 22 - 23

Q 1 | Page 22

Show that the lines  $\frac{x}{1} = \frac{y - 2}{2} = \frac{z + 3}{3} and \frac{x - 2}{2} = \frac{y - 6}{3} = \frac{z - 3}{4}$ intersect and find their point of intersection.

Q 2 | Page 22

Show that the lines $\frac{x - 1}{3} = \frac{y + 1}{2} = \frac{z - 1}{5} and \frac{x + 2}{4} = \frac{y - 1}{3} = \frac{z + 1}{- 2}$  do not intersect.

Q 3 | Page 22

Show that the lines $\frac{x + 1}{3} = \frac{y + 3}{5} = \frac{z + 5}{7} and \frac{x - 2}{1} = \frac{y - 4}{3} = \frac{z - 6}{5}$   intersect. Find their point of intersection.

Q 4 | Page 22

Prove that the lines through A (0, −1, −1) and B (4, 5, 1) intersects the line through C (3, 9, 4) and D (−4, 4, 4). Also, find their point of intersection.

Q 5 | Page 22

Prove that the line $\vec{r} = \left( \hat{i }+ \hat{j }- \hat{k} \right) + \lambda\left( 3 \hat{i} - \hat{j} \right) \text{ and } \vec{r} = \left( 4 \hat{i} - \hat{k} \right) + \mu\left( 2 \hat{i} + 3 \hat{k} \right)$ intersect and find their point of intersection.

Q 6.1 | Page 22

Determine whether the following pair of lines intersect or not:

$\vec{r} = \left( \hat{i} - \hat{j} \right) + \lambda\left( 2 \hat{i} + \hat{k} \right) \text{ and } \vec{r} = \left( 2 \hat{i} - \hat{j} \right) + \mu\left( \hat{i} + \hat{j} - \hat{k} \right)$

Q 6.2 | Page 22

Determine whether the following pair of lines intersect or not:

$\frac{x - 1}{2} = \frac{y + 1}{3} = z \text{ and } \frac{x + 1}{5} = \frac{y - 2}{1}; z = 2$

Q 6.3 | Page 22

Determine whether the following pair of lines intersect or not:

$\frac{x - 1}{3} = \frac{y - 1}{- 1} = \frac{z + 1}{0} and \frac{x - 4}{2} = \frac{y - 0}{0} = \frac{z + 1}{3}$

Q 6.4 | Page 22

Determine whether the following pair of lines intersect or not:

$\frac{x - 5}{4} = \frac{y - 7}{4} = \frac{z + 3}{- 5} and \frac{x - 8}{7} = \frac{y - 4}{1} = \frac{3 - 5}{3}$

Q 7 | Page 23

Show that the lines $\vec{r} = 3 \hat{i} + 2 \hat{j} - 4 \hat{k} + \lambda\left( \hat{i} + 2 \hat{j} + 2 \hat{k} \right) \text{ and } \vec{r} = 5 \hat{i} - 2 \hat{j} + \mu\left( 3 \hat{i} + 2 \hat{j} + 6 \hat{k} \right)$ are intersecting. Hence, find their point of intersection.

#### Pages 29 - 30

Q 1 | Page 29

Find the perpendicular distance of the point (3, −1, 11) from the line $\frac{x}{2} = \frac{y - 2}{- 3} = \frac{z - 3}{4} .$

Q 2 | Page 29

Find the perpendicular distance of the point (1, 0, 0) from the line  $\frac{x - 1}{2} = \frac{y + 1}{- 3} = \frac{z + 10}{8} .$   Also, find the coordinates of the foot of the perpendicular and the equation of the perpendicular.

Q 3 | Page 29

Find the foot of the perpendicular drawn from the point A (1, 0, 3) to the joint of the points B (4, 7, 1) and C (3, 5, 3).

Q 4 | Page 29

A (1, 0, 4), B (0, −11, 3), C (2, −3, 1) are three points and D is the foot of perpendicular from A on BC. Find the coordinates of D

Q 5 | Page 29

Find the foot of perpendicular from the point (2, 3, 4) to the line $\frac{4 - x}{2} = \frac{y}{6} = \frac{1 - z}{3} .$ Also, find the perpendicular distance from the given point to the line.

Q 6 | Page 30

Find the equation of the perpendicular drawn from the point P (2, 4, −1) to the line  $\frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{- 9} .$  Also, write down the coordinates of the foot of the perpendicular from P

Q 7 | Page 30

Find the length of the perpendicular drawn from the point (5, 4, −1) to the line $\vec{r} = \hat{i} + \lambda\left( 2 \hat{i} + 9 \hat{j} + 5 \hat{k} \right) .$

Q 8 | Page 30

Find the foot of the perpendicular drawn from the point  $\hat{i} + 6 \hat{j} + 3 \hat{k}$  to the line  $\vec{r} = \hat{j} + 2 \hat{k} + \lambda\left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) .$  Also, find the length of the perpendicular

Q 9 | Page 30

Find the equation of the perpendicular drawn from the point P (−1, 3, 2) to the line  $\vec{r} = \left( 2 \hat{j} + 3 \hat{k} \right) + \lambda\left( 2 \hat{i} + \hat{j} + 3 \hat{k} \right) .$  Also, find the coordinates of the foot of the perpendicular from P.

Q 10 | Page 30

Find the foot of the perpendicular from (0, 2, 7) on the line $\frac{x + 2}{- 1} = \frac{y - 1}{3} = \frac{z - 3}{- 2} .$

Q 11 | Page 30

Find the foot of the perpendicular from (1, 2, −3) to the line $\frac{x + 1}{2} = \frac{y - 3}{- 2} = \frac{z}{- 1} .$

Q 12 | Page 30

Find the equation of line passing through the points A (0, 6, −9) and B (−3, −6, 3). If D is the foot of perpendicular drawn from a point C (7, 4, −1) on the line AB, then find the coordinates of the point D and the equation of line CD

Q 13 | Page 30

Find the distance of the point (2, 4, −1) from the line  $\frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{- 9}$

Q 14 | Page 30

Find the coordinates of the foot of perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, −1, 3) and C(2, −3, −1).

#### Pages 37 - 38

Q 1.1 | Page 37

Find the shortest distance between the following pairs of lines whose vector equations are: $\vec{r} = 3 \hat{i} + 8 \hat{j} + 3 \hat{k} + \lambda\left( 3 \hat{i} - \hat{j} + \hat{k} \right) \text{ and } \vec{r} = - 3 \hat{i} - 7 \hat{j} + 6 \hat{k} + \mu\left( - 3 \hat{i} + 2 \hat{j} + 4 \hat{k} \right)$

Q 1.2 | Page 37

Find the shortest distance between the following pairs of lines whose vector equations are: $\vec{r} = \left( 3 \hat{i} + 5 \hat{j} + 7 \hat{k} \right) + \lambda\left( \hat{i} - 2 \hat{j} + 7 \hat{k} \right) \text{ and } \vec{r} = - \hat{i} - \hat{j} - \hat{k} + \mu\left( 7 \hat{i} - 6 \hat{j} + \hat{k} \right)$

Q 1.3 | Page 37

Find the shortest distance between the following pairs of lines whose vector equations are: $\vec{r} = \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) + \lambda\left( 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \right) \text{ and } \vec{r} = \left( 2 \hat{i} + 4 \hat{j} + 5 \hat{k} \right) + \mu\left( 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \right)$

Q 1.4 | Page 37

Find the shortest distance between the following pairs of lines whose vector equations are: $\vec{r} = \left( 1 - t \right) \hat{i} + \left( t - 2 \right) \hat{j} + \left( 3 - t \right) \hat{k} \text{ and } \vec{r} = \left( s + 1 \right) \hat{i} + \left( 2s - 1 \right) \hat{j} - \left( 2s + 1 \right) \hat{k}$

Q 1.5 | Page 37

Find the shortest distance between the following pairs of lines whose vector equations are: $\vec{r} = \left( \lambda - 1 \right) \hat{i} + \left( \lambda + 1 \right) \hat{j} - \left( 1 + \lambda \right) \hat{k} \text{ and } \vec{r} = \left( 1 - \mu \right) \hat{i} + \left( 2\mu - 1 \right) \hat{j} + \left( \mu + 2 \right) \hat{k}$

Q 1.6 | Page 37

Find the shortest distance between the following pairs of lines whose vector equations are: $\vec{r} = \left( 2 \hat{i} - \hat{j} - \hat{k} \right) + \lambda\left( 2 \hat{i} - 5 \hat{j} + 2 \hat{k} \right) \text{ and }, \vec{r} = \left( \hat{i} + 2 \hat{j} + \hat{k} \right) + \mu\left( \hat{i} - \hat{j} + \hat{k} \right)$

Q 1.7 | Page 37

Find the shortest distance between the following pairs of lines whose vector are: $\vec{r} = \left( \hat{i} + \hat{j} \right) + \lambda\left( 2 \hat{i} - \hat{j} + \hat{k} \right) \text{ and } , \vec{r} = 2 \hat{i} + \hat{j} - \hat{k} + \mu\left( 3 \hat{i} - 5 \hat{j} + 2 \hat{k} \right)$

Q 1.8 | Page 37

Find the shortest distance between the following pairs of lines whose vector equations are: $\vec{r} = \left( 8 + 3\lambda \right) \hat{i} - \left( 9 + 16\lambda \right) \hat{j} + \left( 10 + 7\lambda \right) \hat{k}$$\vec{r} = 15 \hat{i} + 29 \hat{j} + 5 \hat{k} + \mu\left( 3 \hat{i} + 8 \hat{j} - 5 \hat{k} \right)$

Q 2.1 | Page 38

Find the shortest distance between the following pairs of lines whose cartesian equations are: $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} and \frac{x - 2}{3} = \frac{y - 3}{4} = \frac{z - 5}{5}$

Q 2.2 | Page 38

Find the shortest distance between the following pairs of lines whose cartesian equations are : $\frac{x - 1}{2} = \frac{y + 1}{3} = z \text{ and } \frac{x + 1}{3} = \frac{y - 2}{1}; z = 2$

Q 2.3 | Page 38

Find the shortest distance between the following pairs of lines whose cartesian equations are : $\frac{x - 1}{- 1} = \frac{y + 2}{1} = \frac{z - 3}{- 2} and \frac{x - 1}{1} = \frac{y + 1}{2} = \frac{z + 1}{- 2}$

Q 2.4 | Page 38

Find the shortest distance between the following pairs of lines whose cartesian equations are:  $\frac{x - 3}{1} = \frac{y - 5}{- 2} = \frac{z - 7}{1} and \frac{x + 1}{7} = \frac{y + 1}{- 6} = \frac{z + 1}{1}$

Q 3.1 | Page 38

By computing the shortest distance determine whether the following pairs of lines intersect or not  : $\vec{r} = \left( \hat{i} - \hat{j} \right) + \lambda\left( 2 \hat{i} + \hat{k} \right) \text{ and } \vec{r} = \left( 2 \hat{i} - \hat{j} \right) + \mu\left( \hat{i} + \hat{j} - \hat{k} \right)$

Q 3.2 | Page 38

By computing the shortest distance determine whether the following pairs of lines intersect or not: $\vec{r} = \left( \hat{i} + \hat{j} - \hat{k} \right) + \lambda\left( 3 \hat{i} - \hat{j} \right) \text{ and } \vec{r} = \left( 4 \hat{i} - \hat{k} \right) + \mu\left( 2 \hat{i} + 3 \hat{k} \right)$

Q 3.3 | Page 38

By computing the shortest distance determine whether the following pairs of lines intersect or not: $\frac{x - 1}{2} = \frac{y + 1}{3} = z and \frac{x + 1}{5} = \frac{y - 2}{1}; z = 2$

Q 3.4 | Page 38

By computing the shortest distance determine whether the following pairs of lines intersect or not: $\frac{x - 5}{4} = \frac{y - 7}{- 5} = \frac{z + 3}{- 5} and \frac{x - 8}{7} = \frac{y - 7}{1} = \frac{z - 5}{3}$

Q 4.1 | Page 38

Find the shortest distance between the following pairs of parallel lines whose equations are:  $\vec{r} = \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) + \lambda\left( \hat{i} - \hat{j} + \hat{k} \right) \text{ and } \vec{r} = \left( 2 \hat{i} - \hat{j} - \hat{k} \right) + \mu\left( - \hat{i} + \hat{j} - \hat{k} \right)$

Q 4.2 | Page 38

Find the shortest distance between the following pairs of parallel lines whose equations are: $\vec{r} = \left( \hat{i} + \hat{j} \right) + \lambda\left( 2 \hat{i} - \hat{j} + \hat{k} \right) \text{ and } \vec{r} = \left( 2 \hat{i} + \hat{j} - \hat{k} \right) + \mu\left( 4 \hat{i} - 2 \hat{j} + 2 \hat{k} \right)$

Q 5.1 | Page 38

Find the equations of the lines joining the following pairs of vertices and then find the shortest distance between the lines
(i) (0, 0, 0) and (1, 0, 2)

Q 5.2 | Page 38

Find the equations of the lines joining the following pairs of vertices and then find the shortest distance between the lines

(1, 3, 0) and (0, 3, 0)

Q 6 | Page 38

Write the vector equations of the following lines and hence determine the distance between them  $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z + 4}{6} and \frac{x - 3}{4} = \frac{y - 3}{6} = \frac{z + 5}{12}$

Q 7.1 | Page 38

Find the shortest distance between the lines $\vec{r} = \left( \hat{i} + 2 \hat{j} + \hat{k} \right) + \lambda\left( \hat{i} - \hat{j} + \hat{k} \right) \text{ and } , \vec{r} = 2 \hat{i} - \hat{j} - \hat{k} + \mu\left( 2 \hat{i} + \hat{j} + 2 \hat{k} \right)$

Q 7.2 | Page 38

Find the shortest distance between the lines $\frac{x + 1}{7} = \frac{y + 1}{- 6} = \frac{z + 1}{1} and \frac{x - 3}{1} = \frac{y - 5}{- 2} = \frac{z - 7}{1}$

Q 7.3 | Page 38

Find the shortest distance between the lines $\vec{r} = \hat{i} + 2 \hat{j} + 3 \hat{k} + \lambda\left( \hat{i} - 3 \hat{j} + 2 \hat{k} \right) \text{ and } \vec{r} = 4 \hat{i} + 5 \hat{j} + 6 \hat{k} + \mu\left( 2 \hat{i} + 3 \hat{j} + \hat{k} \right)$

Q 7.4 | Page 38

Find the shortest distance between the lines $\vec{r} = 6 \hat{i} + 2 \hat{j} + 2 \hat{k} + \lambda\left( \hat{i} - 2 \hat{j} + 2 \hat{k} \right) \text{ and } \vec{r} = - 4 \hat{i} - \hat{k} + \mu\left( 3 \hat{i} - 2 \hat{j} - 2 \hat{k} \right)$

Q 8 | Page 38

Find the distance between the lines l1 and l2 given by  $\vec{r} = \hat{i} + 2 \hat{j} - 4 \hat{k} + \lambda\left( 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right) \text{ and } , \vec{r} = 3 \hat{i} + 3 \hat{j} - 5 \hat{k} + \mu\left( 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right)$

#### Pages 41 - 42

Q 1 | Page 41

Write the cartesian and vector equations of X-axis.

Q 2 | Page 41

Write the cartesian and vector equations of Y-axis.

Q 3 | Page 41

Write the cartesian and vector equations of Z-axis.

Q 4 | Page 41

Write the vector equation of a line passing through a point having position vector  $\vec{\alpha}$ and parallel to vector $\vec{\beta}$ .

Q 5 | Page 41

Cartesian equations of a line AB are  $\frac{2x - 1}{2} = \frac{4 - y}{7} = \frac{z + 1}{2} .$   Write the direction ratios of a line parallel to AB.

Q 6 | Page 41

Write the direction cosines of the line whose cartesian equations are 6x − 2 = 3y + 1 = 2z − 4.

Q 7 | Page 41

Write the direction cosines of the line $\frac{x - 2}{2} = \frac{2y - 5}{- 3}, z = 2 .$

Q 8 | Page 41

Write the coordinate axis to which the line $\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 1}{0}$ is perpendicular.

Q 9 | Page 41

Write the angle between the lines $\frac{x - 5}{7} = \frac{y + 2}{- 5} = \frac{z - 2}{1} and \frac{x - 1}{1} = \frac{y}{2} = \frac{z - 1}{3} .$

Q 10 | Page 41

Write the direction cosines of the line whose cartesian equations are 2x = 3y = −z.

Q 11 | Page 41

Write the angle between the lines 2x = 3y = −z and 6x = −y = −4z.

Q 12 | Page 41

Write the value of λ for which the lines  $\frac{x - 3}{- 3} = \frac{y + 2}{2\lambda} = \frac{z + 4}{2} and \frac{x + 1}{3\lambda} = \frac{y - 2}{1} = \frac{z + 6}{- 5}$  are perpendicular to each other.

Q 13 | Page 41

Write the formula for the shortest distance between the lines

$\vec{r} = \vec{a_1} + \lambda \vec{b} \text{ and } \vec{r} = \vec{a_2} + \mu \vec{b} .$

Q 14 | Page 41

Write the condition for the lines  $\vec{r} = \vec{a_1} + \lambda \vec{b_1} \text{ and } \vec{r} = \vec{a_2} + \mu \vec{b_2}$ to be intersecting.

Q 15 | Page 41

The cartesian equations of a line AB are  $\frac{2x - 1}{\sqrt{3}} = \frac{y + 2}{2} = \frac{z - 3}{3} .$   Find the direction cosines of a line parallel to AB

Q 16 | Page 41

If the equations of a line AB are

$\frac{3 - x}{1} = \frac{y + 2}{- 2} = \frac{z - 5}{4},$ write the direction ratios of a line parallel to AB

Q 17 | Page 41

Write the vector equation of a line given by $\frac{x - 5}{3} = \frac{y + 4}{7} = \frac{z - 6}{2} .$

Q 18 | Page 42

The equations of a line are given by $\frac{4 - x}{3} = \frac{y + 3}{3} = \frac{z + 2}{6} .$  Write the direction cosines of a line parallel to this line.

Q 19 | Page 42

Find the Cartesian equations of the line which passes through the point (−2, 4 , −5) and is parallel to the line $\frac{x + 3}{3} = \frac{4 - y}{5} = \frac{z + 8}{6} .$

Q 20 | Page 42

Find the angle between the lines

$\vec{r} = \left( 2 \hat{i} - 5 \hat{j} + \hat{k} \right) + \lambda\left( 3 \hat{i} + 2 \hat{j} + 6 \hat{k} \right)$ and

$\vec{r} = 7 \hat{i} - 6 \hat{k} + \mu\left( \hat{i} + 2 \hat{j} + 2 \hat{k} \right)$

Q 21 | Page 42

Find the angle between the lines 2x=3y=-z and 6x =-y=-4z.

#### Pages 42 - 43

Q 1 | Page 42

The angle between the straight lines $\frac{x + 1}{2} = \frac{y - 2}{5} = \frac{z + 3}{4} and \frac{x - 1}{1} = \frac{y + 2}{2} = \frac{z - 3}{- 3}$

a) 45°

(b) 30°

(c) 60°

(d) 90°

Q 2 | Page 42

The lines  $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ and $\frac{x - 1}{- 2} = \frac{y - 2}{- 4} = \frac{z - 3}{- 6}$ are

(a) coincident

(b) skew

(c) intersecting

(d) parallel

Q 3 | Page 42

The direction ratios of the line perpendicular to the lines $\frac{x - 7}{2} = \frac{y + 17}{- 3} = \frac{z - 6}{1} and, \frac{x + 5}{1} = \frac{y + 3}{2} = \frac{z - 4}{- 2}$

(a) 4, 5, 7

(b) 4, −5, 7

(c) 4, −5, −7

(d) −4, 5, 7

Q 4 | Page 42

The angle between the lines

$\frac{x - 1}{1} = \frac{y - 1}{1} = \frac{z - 1}{2} and, \frac{x - 1}{- \sqrt{3} - 1} = \frac{y - 1}{\sqrt{3} - 1} = \frac{z - 1}{4}$ is

(a) $\cos^{- 1} \left( \frac{1}{65} \right)$

(b) $\frac{\pi}{6}$

(c) $\frac{\pi}{3}$

(d) $\frac{\pi}{4}$

Q 5 | Page 43

The direction ratios of the line x − y + z − 5 = 0 = x − 3y − 6 are proportional to

(a) 3, 1, −2

(b) 2, −4, 1

(c) $\frac{3}{\sqrt{14}}, \frac{1}{\sqrt{14}}, \frac{- 2}{\sqrt{14}}$

(d)  $\frac{2}{\sqrt{41}}, \frac{- 4}{\sqrt{41}}, \frac{1}{\sqrt{41}}$

Q 6 | Page 43

The perpendicular distance of the point P (1, 2, 3) from the line $\frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{- 2}$ is

(a) 7

(b)  5

(c) 0

(d) none of these

Q 7 | Page 43

The equation of the line passing through the points $a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \text{ and } b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}$  is

(a)  $\vec{r} = \left( a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \right) + \lambda \left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right)$

(b)  $\vec{r} = \left( a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \right) - t \left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right)$

(c) $\vec{r} = a_1 \left( 1 - t \right) \hat{i} + a_2 \left( 1 - t \right) \hat{j} + a_3 \left( 1 - t \right) \hat{k} + t \left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right)$

(d) none of these

Q 8 | Page 43

If a line makes angles α, β and γ with the axes respectively, then cos 2 α + cos 2 β + cos 2 γ =

(a) −2

(b) −1

(c) 1

(d) 2

Q 9 | Page 43

If the direction ratios of a line are proportional to 1, −3, 2, then its direction cosines are

(a) $\frac{1}{\sqrt{14}}, - \frac{3}{\sqrt{14}}, \frac{2}{\sqrt{14}}$

(b) $\frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}}$

(c) $- \frac{1}{\sqrt{14}}, \frac{3}{\sqrt{14}}, \frac{2}{\sqrt{14}}$

(d)  $- \frac{1}{\sqrt{14}}, - \frac{2}{\sqrt{14}}, - \frac{3}{\sqrt{14}}$

Q 10 | Page 43

If a line makes angle $\frac{\pi}{3} and \frac{\pi}{4}$  with x-axis and y-axis respectively, then the angle made by the line with z-axis is

(a) π/2

(b) π/3

(c) π/4

(d) 5π/12

Q 11 | Page 43

The projections of a line segment on XY and Z axes are 12, 4 and 3 respectively. The length and direction cosines of the line segment are

(a) $13; \frac{12}{13}, \frac{4}{13}, \frac{3}{13}$

(b)  $19; \frac{12}{19}, \frac{4}{19}, \frac{3}{19}$

(c)  $11; \frac{12}{11}, \frac{14}{11}, \frac{3}{11}$

(d) none of these

Q 12 | Page 43

The lines  $\frac{x}{1} = \frac{y}{2} = \frac{z}{3} and \frac{x - 1}{- 2} = \frac{y - 2}{- 4} = \frac{z - 3}{- 6}$

(a) parallel

(b) intersecting

(c) skew

(d) coincident

Q 13 | Page 43

The straight line $\frac{x - 3}{3} = \frac{y - 2}{1} = \frac{z - 1}{0}$

(a) parallel to x-axis

(b) parallel to y-axis

(c) parallel to z-axis

(d) perpendicular to z-axis

Q 14 | Page 43

The shortest distance between the lines  $\frac{x - 3}{3} = \frac{y - 8}{- 1} = \frac{z - 3}{1} and, \frac{x + 3}{- 3} = \frac{y + 7}{2} = \frac{z - 6}{4}$

(a) $\sqrt{30}$

(b) $2\sqrt{30}$

(c)  $5\sqrt{30}$

(d) $3\sqrt{30}$

#### Pages 4 - 5

Q 1.1 | Page 4

Find the equation of the plane passing through the following points.
(i) (2, 1, 0), (3, −2, −2) and (3, 1, 7)

Q 1.2 | Page 4

Find the equation of the plane passing through the following points.

(ii) (−5, 0, −6), (−3, 10, −9) and (−2, 6, −6)

Q 1.3 | Page 4

Find the equation of the plane passing through the following points.

(iii) (1, 1, 1), (1, −1, 2) and (−2, −2, 2)

Q 1.4 | Page 4

Find the equation of the plane passing through the following points.

(iv) (2, 3, 4), (−3, 5, 1) and (4, −1, 2)

Q 1.5 | Page 4

Find the equation of the plane passing through the following points.

(v) (0, −1, 0), (3, 3, 0) and (1, 1, 1)

Q 2 | Page 5

Show that the four points (0, −1, −1), (4, 5, 1), (3, 9, 4) and (−4, 4, 4) are coplanar and find the equation of the common plane.

Q 3.1 | Page 5

Show that the following points are coplanar.
(i) (0, −1, 0), (2, 1, −1), (1, 1, 1) and (3, 3, 0)

Q 3.2 | Page 5

Show that the following points are coplanar.

(ii) (0, 4, 3), (−1, −5, −3), (−2, −2, 1) and (1, 1, −1)

Q 4 | Page 5

Find the coordinates of the point where the line through (3, -4 , -5 ) and B (2, -3 , 1) crosses the plane passing through three points L(2,2,1), M(3,0,1) and N(4, -1,0 ) . Also, find the ratio in which diveides the line segment AB.

#### Page 7

Q 1 | Page 7

Write the equation of the plane whose intercepts on the coordinate axes are 2, −3 and 4.

Q 2.1 | Page 7

Reduce the equations of the following planes to intercept form and find the intercepts on the coordinate axes.
(i) 4x + 3y − 6z − 12 = 0

Q 2.2 | Page 7

Reduce the equations of the following planes to intercept form and find the intercepts on the coordinate axes.

(ii) 2x + 3y − z = 6

Q 2.3 | Page 7

Reduce the equations of the following planes to intercept form and find the intercepts on the coordinate axes.

(iii) 2x − y + z = 5

Q 3 | Page 7

Find the equation of a plane which meets the axes at AB and C, given that the centroid of the triangle ABC is the point (α, β, γ).

Q 4 | Page 7

Find the equation of the plane passing through the point (2, 4, 6) and making equal intercepts on the coordinate axes.

Q 5 | Page 7

A plane meets the coordinate axes at AB and C, respectively, such that the centroid of triangle ABC is (1, −2, 3). Find the equation of the plane.

#### Page 13

Q 1 | Page 13

Find the vector equation of a plane passing through a point with position vector $2 \hat{i} - \hat{j} + \hat{k}$ and perpendicular to the vector  $4 \hat{i} + 2 \hat{j} - 3 \hat{k} .$

Q 2.1 | Page 13

Find the Cartesian form of the equation of a plane whose vector equation is

(i) $\vec{r} \cdot \left( 12 \hat{i} - 3 \hat{j} + 4 \hat{k} \right) + 5 = 0$

Q 2.2 | Page 13

Find the Cartesian form of the equation of a plane whose vector equation is

(ii)  $\vec{r} \cdot \left( - \hat{i} + \hat{j} + 2 \hat{k} \right) = 9$

Q 3 | Page 13

Find the vector equations of the coordinate planes.

Q 4.1 | Page 13

Find the vector equation of each one of following planes.

(i) 2x − y + 2z = 8

Q 4.2 | Page 13

Find the vector equation of each one of following planes.

(ii) x + y − z = 5

Q 4.3 | Page 13

Find the vector equation of each one of following planes.

(iii) x + y = 3

Q 5 | Page 13

Find the vector and Cartesian equations of a plane passing through the point (1, −1, 1) and normal to the line joining the points (1, 2, 5) and (−1, 3, 1).

Q 6 | Page 13

$\vec{n}$ is a vector of magnitude $\sqrt{3}$ and is equally inclined to an acute angle with the coordinate axes. Find the vector and Cartesian forms of the equation of a plane which passes through (2, 1, −1) and is normal to $\vec{n}$ .

Q 7 | Page 13

The coordinates of the foot of the perpendicular drawn from the origin to a plane are (12, −4, 3). Find the equation of the plane.

Q 8 | Page 13

Find the equation of the plane passing through the point (2, 3, 1), given that the direction ratios of the normal to the plane are proportional to 5, 3, 2.

Q 9 | Page 13

If the axes are rectangular and P is the point (2, 3, −1), find the equation of the plane through P at right angles to OP.

Q 10 | Page 13

Find the intercepts made on the coordinate axes by the plane 2x + y − 2z = 3 and also find the direction cosines of the normal to the plane.

Q 11 | Page 13

A plane passes through the point (1, −2, 5) and is perpendicular to the line joining the origin to the point

$3 \hat{i} + \hat{j} - \hat{k} .$ Find the vector and Cartesian forms of the equation of the plane.

Q 12 | Page 13

Find the equation of the plane that bisects the line segment joining the points (1, 2, 3) and (3, 4, 5) and is at right angle to it.

Q 13.1 | Page 13

Show that the normals to the following pairs of planes are perpendicular to each other.

(i) x − y + z − 2 = 0 and 3x + 2y − z + 4 = 0

Q 13.2 | Page 13

Show that the normals to the following pairs of planes are perpendicular to each other.

$\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + 3 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( 2 \hat{i} - 2 \hat{j} - 2 \hat{k} \right) = 5$

Q 14 | Page 13

Show that the normal vector to the plane 2x + 2y + 2z = 3 is equally inclined to the coordinate axes.

#### Page 19

Q 1 | Page 19

Find the vector equation of a plane which is at a distance of 3 units from the origin and has $\hat{k}$ as the unit vector normal to it.

Q 2 | Page 19

Find the vector equation of a plane which is at a distance of 5 units from the origin and which is normal to the vector  $\hat{i} - 2 \hat{j} - 2 \hat{k} .$

Q 3 | Page 19

Reduce the equation 2x − 3y − 6z = 14 to the normal form and, hence, find the length of the perpendicular from the origin to the plane. Also, find the direction cosines of the normal to the plane.

Q 4 | Page 19

Reduce the equation $\vec{r} \cdot \left( \hat{i} - 2 \hat{j} + 2 \hat{k} \right) + 6 = 0$ to normal form and, hence, find the length of the perpendicular from the origin to the plane.

Q 5 | Page 19

Write the normal form of the equation of the plane 2x − 3y + 6z + 14 = 0.

Q 6 | Page 19

The direction ratios of the perpendicular from the origin to a plane are 12, −3, 4 and the length of the perpendicular is 5. Find the equation of the plane.

Q 7 | Page 19

Find a unit normal vector to the plane x + 2y + 3z − 6 = 0.

Q 8 | Page 19

Find the equation of a plane which is at a distance of $3\sqrt{3}$  units from the origin and the normal to which is equally inclined to the coordinate axes.

Q 9 | Page 19

find the equation of the plane passing through the point (1, 2, 1) and perpendicular to the line joining the points (1, 4, 2) and (2, 3, 5). Find also the perpendicular distance of the origin from this plane

Q 10 | Page 19

Find the vector equation of the plane which is at a distance of $\frac{6}{\sqrt{29}}$ from the origin and its normal vector from the origin is  $2 \hat{i} - 3 \hat{j} + 4 \hat{k} .$ Also, find its Cartesian form.

Q 11 | Page 19

Find the distance of the plane 2x − 3y + 4z − 6 = 0 from the origin.

#### Pages 22 - 23

Q 1 | Page 22

Find the vector equation of the plane passing through the points (1, 1, 1), (1, −1, 1) and (−7, −3, −5).

Q 2 | Page 23

Find the vector equation of the plane passing through the points P (2, 5, −3), Q (−2, −3, 5) and R (5, 3, −3).

Q 3 | Page 23

Find the vector equation of the plane passing through points A (a, 0, 0), B (0, b, 0) and C(0, 0, c). Reduce it to normal form. If plane ABC is at a distance p from the origin, prove that $\frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} .$

Q 4 | Page 23

Find the vector equation of the plane passing through the points (1, 1, −1), (6, 4, −5) and (−4, −2, 3).

Q 5 | Page 23

Find the vector equation of the plane passing through the points $3 \hat{i} + 4 \hat{j} + 2 \hat{k} , 2 \hat{i} - 2 \hat{j} - \hat{k} \text{ and } 7 \hat{i} + 6 \hat{k} .$

#### Page 29

Q 1.1 | Page 29

Find the angle between the given planes. $\vec{r} \cdot \left( 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \right) = 1 \text{ and } \vec{r} \cdot \left( - \hat{i} + \hat{j} \right) = 4$

Q 1.2 | Page 29

Find the angle between the given planes. $\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) = 6 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + 6 \hat{j} - 2 \hat{k} \right) = 9$

Q 1.3 | Page 29
Find the angle between the given planes.
$\vec{r} \cdot \left( 2 \hat{i} + 3 \hat{j} - 6 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( \hat{i} - 2 \hat{j} + 2 \hat{k} \right) = 9$

Q 2.1 | Page 29

Find the angle between the planes.

(i) 2x − y + z = 4 and x + y + 2z = 3

Q 2.2 | Page 29

Find the angle between the planes.

(ii) x + y − 2z = 3 and 2x − 2y + z = 5

Q 2.3 | Page 29

Find the angle between the planes.

(iii) x − y + z = 5 and x + 2y + z = 9

Q 2.4 | Page 29

Find the angle between the planes.
(iv) 2x − 3y + 4z = 1 and − x + y = 4

Q 2.5 | Page 29

Find the angle between the planes.

(v) 2x + y − 2z = 5 and 3x − 6y − 2z = 7

Q 3.1 | Page 29

Show that the following planes are at right angles.

$\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( - \hat{i} - \hat{j} + \hat{k} \right) = 3$

Q 3.2 | Page 29

Show that the following planes are at right angles.

(ii) x − 2y + 4z = 10 and 18x + 17y + 4z = 49

Q 4.1 | Page 29

Determine the value of λ for which the following planes are perpendicular to each other.

$\vec{r} \cdot \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) = 7 \text{ and } \vec{r} \cdot \left( \lambda \hat{i} + 2 \hat{j} - 7 \hat{k} \right) = 26$

Q 4.2 | Page 29

Determine the value of λ for which the following planes are perpendicular to each other.

(ii) 2x − 4y + 3z = 5 and x + 2y + λz = 5

Q 4.3 | Page 29

Determine the value of λ for which the following planes are perpendicular to each other.

(iii) 3x − 6y − 2z = 7 and 2x + y − λz = 5

Q 5 | Page 29

Find the equation of a plane passing through the point (−1, −1, 2) and perpendicular to the planes 3x + 2y − 3z = 1 and 5x − 4y + z = 5.

Q 6 | Page 29

Obtain the equation of the plane passing through the point (1, −3, −2) and perpendicular to the planes x + 2y + 2z = 5 and 3x + 3y + 2z = 8.

Q 7 | Page 29

Find the equation of the plane passing through the origin and perpendicular to each of the planes x + 2y − z = 1 and 3x − 4y + z = 5.

Q 8 | Page 29

Find the equation of the plane passing through the points (1, −1, 2) and (2, −2, 2) and which is perpendicular to the plane 6x − 2y + 2z = 9.

Q 9 | Page 29

Find the equation of the plane passing through the points (2, 2, 1) and (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 1.

Q 10 | Page 29

Find the equation of the plane passing through the points whose coordinates are (−1, 1, 1) and (1, −1, 1) and perpendicular to the plane x + 2y + 2z = 5.

Q 11 | Page 29

Find the equation of the plane with intercept 3 on the y-axis and parallel to the ZOX plane.

Q 12 | Page 29

Find the equation of the plane that contains the point (1, −1, 2) and is perpendicular to each of the planes 2x + 3y − 2z = 5 and x + 2y − 3z = 8.

Q 13 | Page 29

Find the equation of the plane passing through (abc) and parallel to the plane  $\vec{r} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 2 .$

Q 14 | Page 29

Find the equation of the plane passing through the point (−1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.

Q 15 | Page 29

Find the vector equation of the plane through the points (2, 1, −1) and (−1, 3, 4) and perpendicular to the plane x − 2y + 4z = 10

#### Page 33

Q 1.1 | Page 33

Find the vector equations of the following planes in scalar product form  $\left( \vec{r} \cdot \vec{n} = d \right):$ $\vec{r} = \left( 2 \hat{i} - \hat{k} \right) + \lambda \hat{i} + \mu\left( \hat{i} - 2 \hat{j} - \hat{k} \right)$

Q 1.2 | Page 33

Find the vector equations of the following planes in scalar product form  $\left( \vec{r} \cdot \vec{n} = d \right):$ $\vec{r} = \left( 1 + s - t \right) \hat{t} + \left( 2 - s \right) \hat{j} + \left( 3 - 2s + 2t \right) \hat{k}$

Q 1.3 | Page 33

Find the vector equations of the following planes in scalar product form  $\left( \vec{r} \cdot \vec{n} = d \right):$$\vec{r} = \left( \hat{i} + \hat{j} \right) + \lambda\left( \hat{i} + 2 \hat{j} - \hat{k} \right) + \mu\left( - \hat{i} + \hat{j} - 2 \hat{k} \right)$

Q 1.4 | Page 33

Find the vector equations of the following planes in scalar product form  $\left( \vec{r} \cdot \vec{n} = d \right):$$\vec{r} = \hat{i} - \hat{j} + \lambda\left( \hat{i} + \hat{j} + \hat{k} \right) + \mu\left( 4 \hat{i} - 2 \hat{j} + 3 \hat{k} \right)$

Q 2.1 | Page 33

Find the Cartesian forms of the equations of the following planes. $\vec{r} = \left( \hat{i} - \hat{j} \right) + s\left( - \hat{i} + \hat{j} + 2 \hat{k} \right) + t\left( \hat{i} + 2 \hat{j} + \hat{k} \right)$

Q 2.2 | Page 33

Find the Cartesian forms of the equations of the following planes.

$\vec{r} = \left( 1 + s + t \right) \hat{i} + \left( 2 - s + t \right) \hat{i} + \left( 3 - 2s + 2t \right) \hat{k}$

Q 3.1 | Page 33

Find the vector equation of the following planes in non-parametric form. $\vec{r} = \left( \lambda - 2\mu \right) \hat{i} + \left( 3 - \mu \right) \hat{j} + \left( 2\lambda + \mu \right) \hat{k}$

Q 3.2 | Page 33

Find the vector equation of the following planes in non-parametric form. $\vec{r} = \left( 2 \hat{i} + 2 \hat{j} - \hat{k} \right) + \lambda\left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) + \mu\left( 5 \hat{i} - 2 \hat{j} + 7 \hat{k} \right)$

#### Page 39

Q 1 | Page 39

Find the equation of the plane which is parallel to 2x − 3y + z = 0 and which passes through (1, −1, 2).

Q 2 | Page 39

Find the equation of the plane through (3, 4, −1) which is parallel to the plane $\vec{r} \cdot \left( 2 \hat{i} - 3 \hat{j} + 5 \hat{k} \right) + 2 = 0 .$

Q 3 | Page 39

Find the equation of the plane passing through the line of intersection of the planes 2x − 7y + 4z − 3 = 0, 3x − 5y + 4z + 11 = 0 and the point (−2, 1, 3).

Q 4 | Page 39

Find the equation of the plane through the point $2 \hat{i} + \hat{j} - \hat{k}$ and passing through the line of intersection of the planes $\vec{r} \cdot \left( \hat{i} + 3 \hat{j} - \hat{k} \right) = 0 \text{ and } \vec{r} \cdot \left( \hat{j} + 2 \hat{k} \right) = 0 .$

Q 5 | Page 39

Find the equation of the plane passing through the line of intersection of the planes 2x − y = 0 and 3z − y = 0 and perpendicular to the plane 4x + 5y − 3z = 8

Q 6 | Page 39

Find the equation of the plane which contains the line of intersection of the planes x + 2y + 3z − 4 = 0 and 2x + y − z + 5 = 0 and which is perpendicular to the plane 5x + 3y − 6z+ 8 = 0.

Q 7 | Page 39

Find the equation of the plane through the line of intersection of the planes x + 2y + 3z + 4 = 0 and x − y + z + 3 = 0 and passing through the origin.

Q 8 | Page 39

Find the vector equation (in scalar product form) of the plane containing the line of intersection of the planes x − 3y + 2z − 5 = 0 and 2x − y + 3z − 1 = 0 and passing through (1, −2, 3).

Q 9 | Page 39

Find the equation of the plane that is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z − 4 = 0, 2x + y − z + 5 = 0.

Q 10 | Page 39

Find the equation of the plane through the line of intersection of the planes  $\vec{r} \cdot \left( \hat{i} + 3 \hat{j} \right) + 6 = 0 \text{ and } \vec{r} \cdot \left( 3 \hat{i} - \hat{j} - 4 \hat{k} \right) = 0,$ which is at a unit distance from the origin.

Q 11 | Page 39

Find the equation of the plane passing through the intersection of the planes 2x + 3y − z+ 1 = 0 and x + y − 2z + 3 = 0 and perpendicular to the plane 3x − y − 2z − 4 = 0.

Q 12 | Page 39

Find the equation of the plane that contains the line of intersection of the planes  $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) - 4 = 0 \text{ and } \vec{r} \cdot \left( 2 \hat{i} + \hat{j} - \hat{k} \right) + 5 = 0$ and which is perpendicular  to the plane $\vec{r} \cdot \left( 5 \hat{i} + 3 \hat{j} - 6 \hat{k} \right) + 8 = 0 .$

Q 13 | Page 39

Find the equation of the plane passing through (abc) and parallel to the plane $\vec{r} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 2 .$

Q 14 | Page 39

Find the equation of the plane passing through the intersection of the planes  $\vec{r} \cdot \left( 2 \hat{i} + \hat{j} + 3 \hat{k} \right) = 7, \vec{r} \cdot \left( 2 \hat{i} + 5 \hat{j} + 3 \hat{k} \right) = 9$ and the point (2, 1, 3).

#### Page 49

Q 1 | Page 49

Find the distance of the point  $2 \hat{i} - \hat{j} - 4 \hat{k}$  from the plane  $\vec{r} \cdot \left( 3 \hat{i} - 4 \hat{j} + 12 \hat{k} \right) - 9 = 0 .$

Q 2 | Page 49

Show that the points $\hat{i} - \hat{j} + 3 \hat{k} \text{ and } 3 \hat{i} + 3 \hat{j} + 3 \hat{k}$ are equidistant from the plane $\vec{r} \cdot \left( 5 \hat{i} + 2 \hat{j} - 7 \hat{k} \right) + 9 = 0 .$

Q 3 | Page 49

Find the distance of the point (2, 3, −5) from the plane x + 2y − 2z − 9 = 0.

Q 4 | Page 49

Find the equations of the planes parallel to the plane x + 2y − 2z + 8 = 0 that are at a distance of 2 units from the point (2, 1, 1).

Q 5 | Page 49

Show that the points (1, 1, 1) and (−3, 0, 1) are equidistant from the plane 3x + 4y − 12z + 13 = 0.

Q 6 | Page 49

Find the equations of the planes parallel to the plane x − 2y + 2z − 3 = 0 and which are at a unit distance from the point (1, 1, 1).

Q 7 | Page 49

Find the distance of the point (2, 3, 5) from the xy - plane.

Q 8 | Page 49

Find the distance of the point (3, 3, 3) from the plane $\vec{r} \cdot \left( 5 \hat{i} + 2 \hat{j} - 7k \right) + 9 = 0$

Q 9 | Page 49

If the product of the distances of the point (1, 1, 1) from the origin and the plane x − y + z+ λ = 0 be 5, find the value of λ.

Q 10 | Page 49

Find an equation for the set of all points that are equidistant from the planes 3x − 4y + 12z = 6 and 4x + 3z = 7.

Q 11 | Page 49

Find the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, −3), B(−2, −3, 5) and C(5, 3, −3).

Q 12 | Page 49

A plane makes intercepts −6, 3, 4 respectively on the coordinate axes. Find the length of the perpendicular from the origin on it.

Q 13 | Page 49

Find the distance of the point (1, -2, 4) from plane passing throuhg the point (1, 2, 2) and perpendicular of the planes  $x - y +$ 2  z = 3 and 2 x - 2  y +z + 12 = 0.

#### Page 51

Q 1 | Page 51

Find the distance between the parallel planes 2x − y + 3z − 4 = 0 and 6x − 3y + 9z + 13 = 0.

Q 2 | Page 51

Find the equation of the plane which passes through the point (3, 4, −1) and is parallel to the plane 2x − 3y + 5z + 7 = 0. Also, find the distance between the two planes.

Q 3 | Page 51

Find the equation of the plane mid-parallel to the planes 2x − 2y + z + 3 = 0 and 2x − 2y + z + 9 = 0.

Q 4 | Page 51

Find the distance between the planes $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) + 7 = 0 \text{ and } \vec{r} \cdot \left( 2 \hat{i} + 4 \hat{j} + 6 \hat{k} \right) + 7 = 0 .$

#### Pages 61 - 62

Q 1 | Page 61

Find the angle between the line $\vec{r} = \left( 2 \hat{i}+ 3 \hat {j} + 9 \hat{k} \right) + \lambda\left( 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \right)$  and the plane  $\vec{r} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 5 .$

Q 2 | Page 61

Find the angle between the line $\frac{x - 1}{1} = \frac{y - 2}{- 1} = \frac{z + 1}{1}$  and the plane 2x + y − z = 4.

Q 3 | Page 61

Find the angle between the line joining the points (3, −4, −2) and (12, 2, 0) and the plane 3x − y + z = 1.

Q 4 | Page 61

The line  $\vec{r} = \hat{i} + \lambda\left( 2 \hat{i} - m \hat{j} - 3 \hat{k} \right)$  is parallel to the plane  $\vec{r} \cdot \left( m \hat{i} + 3 \hat{j} + \hat{k} \right) = 4 .$ Find m

Q 5 | Page 61

Show that the line whose vector equation is $\vec{r} = 2 \hat{i} + 5 \hat{j} + 7 \hat{k}+ \lambda\left( \hat{i} + 3 \hat{j} + 4 \hat{k} \right)$ is parallel to the plane whose vector  $\vec{r} \cdot \left( \hat{i} + \hat{j} - \hat{k} \right) = 7 .$  Also, find the distance between them.

Q 6 | Page 61

Find the vector equation of the line through the origin which is perpendicular to the plane  $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) = 3 .$

Q 7 | Page 61

Find the equation of the plane through (2, 3, −4) and (1, −1, 3) and parallel to x-axis.

Q 8 | Page 61

Find the equation of a plane passing through the points (0, 0, 0) and (3, −1, 2) and parallel to the line $\frac{x - 4}{1} = \frac{y + 3}{- 4} = \frac{z + 1}{7} .$

Q 9 | Page 61

Find the vector and Cartesian equations of the line passing through (1, 2, 3) and parallel to the planes $\vec{r} \cdot \left( \hat{i} - \hat{j} + 2 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + \hat{j} + 2 \hat{k} \right) = 6$

Q 10 | Page 61

Prove that the line of section of the planes 5x + 2y − 4z + 2 = 0 and 2x + 8y + 2z − 1 = 0 is parallel to the plane 4x − 2y − 5z − 2 = 0.

Q 11 | Page 61

Find the vector equation of the line passing through the point (1, −1, 2) and perpendicular to the plane 2x − y + 3z − 5 = 0.

Q 12 | Page 61

Find the equation of the plane through the points (2, 2, −1) and (3, 4, 2) and parallel to the line whose direction ratios are 7, 0, 6.

Q 13 | Page 61

Find the angle between the line $\frac{x - 2}{3} = \frac{y + 1}{- 1} = \frac{z - 3}{2}$ and the plane

3x + 4y + z + 5 = 0.

Q 14 | Page 61

Find the equation of the plane passing through the intersection of the planes x − 2y + z = 1 and 2x + y + z = 8 and parallel to the line with direction ratios proportional to 1, 2, 1. Also, find the perpendicular distance of (1, 1, 1) from this plane

Q 15 | Page 61

State when the line $\vec{r} = \vec{a} + \lambda \vec{b}$  is parallel to the plane  $\vec{r} \cdot \vec{n} = d .$Show that the line  $\vec{r} = \hat{i} + \hat{j} + \lambda\left( 3 \hat{i} - \hat{j} + 2 \hat{k} \right)$  is parallel to the plane  $\vec{r} \cdot \left( 2 \hat{j} + \hat{k} \right) = 3 .$   Also, find the distance between the line and the plane.

Q 16 | Page 61

Show that the plane whose vector equation is $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} - \hat{k} \right) = 1$ and the line whose vector equation is  $\vec{r} = \left( - \hat{i} + \hat{j} + \hat{k} \right) + \lambda\left( 2 \hat{i} + \hat{j} + 4 \hat{k} \right)$   are parallel. Also, find the distance between them.

Q 17 | Page 61

Find the equation of the plane through the intersection of the planes 3x − 4y + 5z = 10 and 2x + 2y − 3z = 4 and parallel to the line x = 2y = 3z.

Q 18 | Page 62

Find the vector and Cartesian forms of the equation of the plane passing through the point (1, 2, −4) and parallel to the lines $\vec{r} = \left( \hat{i} + 2 \hat{j} - 4 \hat{k} \right) + \lambda\left( 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right)$ and $\vec{r} = \left( \hat{i} - 3 \hat{j} + 5 \hat{k} \right) + \mu\left( \hat{i} + \hat{j} - \hat{k} \right)$ Also, find the distance of the point (9, −8, −10) from the plane thus obtained.

Q 19 | Page 62

Find the equation of the plane passing through the points (3, 4, 1) and (0, 1, 0) and parallel to the line

$\frac{x + 3}{2} = \frac{y - 3}{7} = \frac{z - 2}{5} .$

Q 20 | Page 62

Find the coordinates of the point where the line  $\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{2}$   intersects the plane x − y + z − 5 = 0. Also, find the angle between the line and the plane.

Q 21 | Page 62

Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} - 5 \hat{k} \right) + 9 = 0 .$

Q 22 | Page 62

Find the angle between the line

$\frac{x + 1}{2} = \frac{y}{3} = \frac{z - 3}{6}$  and the plane 10x + 2y − 11z = 3.

Q 23 | Page 62

Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes  $\vec{r} \cdot \left( \hat{i} - \hat{j} + 2 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + \hat{j} + \hat{k} \right) = 6 .$

Q 24 | Page 62

Find the value of λ such that the line $\frac{x - 2}{6} = \frac{y - 1}{\lambda} = \frac{z + 5}{- 4}$  is perpendicular to the plane 3x − y − 2z = 7.

Q 25 | Page 62

Find the equation of the plane passing through the points (−1, 2, 0), (2, 2, −1) and parallel to the line $\frac{x - 1}{1} = \frac{2y + 1}{2} = \frac{z + 1}{- 1}$

#### Page 65

Q 1.1 | Page 65

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the   yz - plane .

Q 1.2 | Page 65

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the  zx - plane .

Q 2 | Page 65

Find the coordinates of the point where the line through (3, −4, −5) and (2, −3, 1) crosses the plane 2x + y + z = 7.

Q 3 | Page 65

Find the distance of the point (−1, −5, −10) from the point of intersection of the line $\vec{r} = \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) + \lambda\left( 3 \hat{i}+ 4 \hat{j} + 2 \hat{k} \right)$ and the plane  $\vec{r} . \left( \hat{i} - \hat{j} + \hat{k} \right) = 5 .$

Q 4 | Page 65

Find the distance of the point (2, 12, 5) from the point of intersection of the line $\vec{r} = 2 \hat{i} - 4 \hat{j}+ 2 \hat{k} + \lambda\left( 3 \hat{i} + 4 \hat{j} + 2 \hat{k} \right)$ and $\vec{r} . \left( \hat{i} - 2 \hat{j} + \hat{k} \right) = 0$

Q 5 | Page 65

Find the distance of the point P(−1, −5, −10) from the point of intersection of the line joining the points A(2, −1, 2) and B(5, 3, 4) with the plane  $x - y + z = 5$ .

Q 6 | Page 65

Find the distance of the point P(3, 4, 4) from the point, where the line joining the points A(3, −4, −5) and B(2, −3, 1) intersects the plane 2x + y + z = 7.

Q 7 | Page 65

Find the distance of the point (1, -5, 9) from the plane

$x - y + z =$ 5  measured along the line $x = y = z$  .

#### Pages 73 - 74

Q 1 | Page 73

Show that the lines $\vec{r} = \left( 2 \hat{j} - 3 \hat{k} \right) + \lambda\left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) \text{ and } \vec{r} = \left( 2 \hat{i} + 6 \hat{j} + 3 \hat{k} \right) + \mu\left( 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \right)$  are coplanar. Also, find the equation of the plane containing them.

Q 2 | Page 74

Show that the lines $\frac{x + 1}{- 3} = \frac{y - 3}{2} = \frac{z + 2}{1} and \frac{x}{1} = \frac{y - 7}{- 3} = \frac{z + 7}{2}$  are coplanar. Also, find the equation of the plane containing them.

Q 3 | Page 74

Find the equation of the plane containing the line $\frac{x + 1}{- 3} = \frac{y - 3}{2} = \frac{z + 2}{1}$  and the point (0, 7, −7) and show that the line  $\frac{x}{1} = \frac{y - 7}{- 3} = \frac{z + 7}{2}$ also lies in the same plane.

Q 4 | Page 74

Find the equation of the plane which contains two parallel lines$\frac{x - 4}{1} = \frac{y - 3}{- 4} = \frac{z - 2}{5} and \frac{x - 3}{1} = \frac{y + 2}{- 4} = \frac{z}{5} .$

Q 5 | Page 74

Show that the lines  $\frac{x + 4}{3} = \frac{y + 6}{5} = \frac{z - 1}{- 2}$ and 3x − 2y + z + 5 = 0 = 2x + 3y + 4z − 4 intersect. Find the equation of the plane in which they lie and also their point of intersection.

Q 6 | Page 74

Show that the plane whose vector equation is $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} - \hat{k} \right) = 3$ contains the line whose vector equation is $\vec{r} = \hat{i} + \hat{j} + \lambda\left( 2 \hat{i} + \hat{j} + 4 \hat{k} \right) .$

Q 7 | Page 74

Find the equation of the plane determined by the intersection of the lines $\frac{x + 3}{3} = \frac{y}{- 2} = \frac{z - 7}{6} and \frac{x + 6}{1} = \frac{y + 5}{- 3} = \frac{z - 1}{2}$

Q 8 | Page 74

Find the vector equation of the plane passing through the points (3, 4, 2) and (7, 0, 6) and perpendicular to the plane 2x − 5y − 15 = 0. Also, show that the plane thus obtained contains the line $\vec{r} = \hat{i} + 3 \hat{j} - 2 \hat{k} + \lambda\left( \hat{i} - \hat{j} + \hat{k} \right) .$

Q 9 | Page 74

If the lines  $\frac{x - 1}{- 3} = \frac{y - 2}{- 2k} = \frac{z - 3}{2} and \frac{x - 1}{k} = \frac{y - 2}{1} = \frac{z - 3}{5}$ are perpendicular, find the value of and, hence, find the equation of the plane containing these lines.

Q 10 | Page 74

Find the coordinates of the point where the line $\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{2}$ intersect the plane x − y + z − 5 = 0. Also, find the angle between the line and the plane.

Q 11 | Page 74

Find the vector equation of the plane passing through three points with position vectors  $\hat{i} + \hat{j} - 2 \hat{k} , 2 \hat{i} - \hat{j} + \hat{k} \text{ and } \hat{i} + 2 \hat{j} + \hat{k} .$  Also, find the coordinates of the point of intersection of this plane and the line  $\vec{r} = 3 \hat{i} - \hat{j} - \hat{k} + \lambda\left( 2 \hat{i} - 2 \hat{j} + \hat{k} \right) .$

Q 12 | Page 74

Show that the lines  $\frac{5 - x}{- 4} = \frac{y - 7}{4} = \frac{z + 3}{- 5}$ and  $\frac{x - 8}{7} = \frac{2y - 8}{2} = \frac{z - 5}{3}$ are coplanar.

Q 13 | Page 74

Find the equation of a plane which passes through the point (3, 2, 0) and contains the line  $\frac{x - 3}{1} = \frac{y - 6}{5} = \frac{z - 4}{4}$ .

Q 14 | Page 74

Show that the lines  $\frac{x + 3}{- 3} = \frac{y - 1}{1} = \frac{z - 5}{5}$ and  $\frac{x + 1}{- 1} = \frac{y - 2}{2} = \frac{z - 5}{5}$  are coplanar. Hence, find the equation of the plane containing these lines.

Q 15 | Page 74

If the line $\frac{x - 3}{2} = \frac{y + 2}{- 1} = \frac{z + 4}{3}$  lies in the plane  $lx + my - z =$   then find the value of  $l^2 + m^2$ .

Q 16 | Page 74

Find the values of  $\lambda$ for which the lines

$\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z + 3}{\lambda^2}$
and  $\frac{x - 3}{1} = \frac{y - 2}{\lambda^2} = \frac{z - 1}{2}$  are coplanar .
Q 17 | Page 74

If the lines  $x =$  5 ,  $\frac{y}{3 - \alpha} = \frac{z}{- 2}$ and   $x = \alpha$ $\frac{y}{- 1} = \frac{z}{2 - \alpha}$ are coplanar, find the values of  $\alpha$.

Q 18 | Page 74

If the straight lines  $\frac{x - 1}{2} = \frac{y + 1}{k} = \frac{z}{2}$ and $\frac{x + 1}{2} = \frac{y + 1}{2} = \frac{z}{k}$ are coplanar, find the equations of the planes containing them.

#### Page 77

Q 1 | Page 77

Find the shortest distance between the lines

$\frac{x - 2}{- 1} = \frac{y - 5}{2} = \frac{z - 0}{3} \text{ and } \frac{x - 0}{2} = \frac{y + 1}{- 1} = \frac{z - 1}{2} .$

Q 2 | Page 77

Find the shortest distance between the lines

$\frac{x + 1}{7} = \frac{y + 1}{- 6} = \frac{z + 1}{1} \text{ and } \frac{x - 3}{1} = \frac{y - 5}{- 2} = \frac{z - 7}{1} .$

Q 3 | Page 77

Find the shortest distance between the lines

$\frac{x - 1}{2} = \frac{y - 3}{4} = \frac{z + 2}{1}$ and
$3x - y - 2z + 4 = 0 = 2x + y + z + 1$

#### Pages 81 - 82

Q 1 | Page 81

Find the image of the point (0, 0, 0) in the plane 3x + 4y − 6z + 1 = 0.

Q 2 | Page 81

Find the reflection of the point (1, 2, −1) in the plane 3x − 5y + 4z = 5.

Q 3 | Page 81

Find the coordinates of the foot of the perpendicular drawn from the point (5, 4, 2) to the line $\frac{x + 1}{2} = \frac{y - 3}{3} = \frac{z - 1}{- 1} .$

Hence, or otherwise, deduce the length of the perpendicular.

Q 4 | Page 81

Find the image of the point with position vector$3 \hat{i} + \hat{j} + 2 \hat{k}$  in the plane$\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + \hat{k} \right) = 4 .$  Also, find the position vectors of the foot of the perpendicular and the equation of the perpendicular line through $3 \hat{i} + \hat{j} + 2 \hat{k} .$

Q 5 | Page 81

Find the coordinates of the foot of the perpendicular from the point (1, 1, 2) to the plane 2x − 2y + 4z + 5 = 0. Also, find the length of the perpendicular.

Q 6 | Page 82

Find the distance of the point (1, −2, 3) from the plane x − y + z = 5 measured along a line parallel to  $\frac{x}{2} = \frac{y}{3} = \frac{z}{- 6} .$

Q 7 | Page 82

Find the coordinates of the foot of the perpendicular from the point (2, 3, 7) to the plane 3x − y − z = 7. Also, find the length of the perpendicular.

Q 8 | Page 82

Find the image of the point (1, 3, 4) in the plane 2x − y + z + 3 = 0.

Q 9 | Page 82

Find the distance of the point with position vector

$- \hat{i} - 5 \hat{j} - 10 \hat{k}$  from the point of intersection of the line $\vec{r} = \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) + \lambda\left( 3 \hat{i} + 4 \hat{j} + 12 \hat{k} \right)$  with the plane $\vec{r} \cdot \left( \hat{i} - \hat{j}+ \hat{k} \right) = 5 .$

Q 10 | Page 82

Find the length and the foot of the perpendicular from the point (1, 1, 2) to the plane $\vec{r} \cdot \left( \hat{i} - 2 \hat{j} + 4 \hat{k} \right) + 5 = 0 .$

Q 11 | Page 82

Find the coordinates of the foot of the perpendicular and the perpendicular distance of the  point P (3, 2, 1) from the plane 2x − y + z + 1 = 0. Also, find the image of the point in the plane.

Q 12 | Page 82

Find the direction cosines of the unit vector perpendicular to the plane  $\vec{r} \cdot \left( 6 \hat{i} - 3 \hat{j} - 2 \hat{k} \right) + 1 = 0$ passing through the origin.

Q 13 | Page 82

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x − 3y + 4z − 6 = 0.

Q 14 | Page 82

Find the length and the foot of perpendicular from the point $\left( 1, \frac{3}{2}, 2 \right)$  to the plane $2x - 2y + 4z + 5 = 0$ .

Q 15 | Page 82

Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector $2 \hat{i} + 3 \hat{j} + 4 \hat{k}$ to the plane  $\vec{r} . \left( 2 \hat{i} + \hat{j} + 3 \hat{k} \right) - 26 = 0$ Also find image of P in the plane.

#### Pages 83 - 84

Q 1 | Page 83

Write the equation of the plane parallel to XOY- plane and passing through the point (2, −3, 5).

Q 2 | Page 83

Write the equation of the plane parallel to the YOZ- plane and passing through (−4, 1, 0).

Q 3 | Page 83

Write the equation of the plane passing through points (a, 0, 0), (0, b, 0) and (0, 0, c).

Q 4 | Page 83

Write the general equation of a plane parallel to X-axis.

Q 5 | Page 83

Write the value of k for which the planes x − 2y + kz = 4 and 2x + 5y − z = 9 are perpendicular.

Q 6 | Page 83

Write the intercepts made by the plane 2x − 3y + 4z = 12 on the coordinate axes.

Q 7 | Page 83

Write the ratio in which the plane 4x + 5y − 3z = 8 divides the line segment joining the points (−2, 1, 5) and (3, 3, 2).

Q 8 | Page 83

Write the distance between the parallel planes 2x − y + 3z = 4 and 2x − y + 3z = 18.

Q 9 | Page 83

Write the plane  $\vec{r} \cdot \left( 2 \hat{i} + 3 \hat{j} - 6 \hat{k} \right) = 14$  in normal form.

Q 10 | Page 83

Write the distance of the plane  $\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) = 12$ from the origin.

Q 11 | Page 83

Write the equation of the plane  $\vec{r} = \vec{a} + \lambda \vec{b} + \mu \vec{c}$   in scalar product form.

Q 12 | Page 83

Write a vector normal to the plane  $\vec{r} = l \vec{b} + m \vec{c} .$

Q 13 | Page 83

Write the equation of the plane passing through (2, −1, 1) and parallel to the plane 3x + 2y −z = 7.

Q 14 | Page 83

Write the equation of the plane containing the lines $\vec{r} = \vec{a} + \lambda \vec{b} \text{ and } \vec{r} = \vec{a} + \mu \vec{c} .$

Q 15 | Page 83

Write the position vector of the point where the line $\vec{r} = \vec{a} + \lambda \vec{b}$ meets the plane  $\vec{r} . \vec{n} = 0 .$

Q 16 | Page 83

Write the value of k for which the line $\frac{x - 1}{2} = \frac{y - 1}{3} = \frac{z - 1}{k}$  is perpendicular to the normal to the plane  $\vec{r} \cdot \left( 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \right) = 4 .$

Q 17 | Page 84

Write the angle between the line $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z + 3}{- 2}$  and the plane x + y + 4 = 0.

Q 18 | Page 84

Write the intercept cut off by the plane 2x + y − z = 5 on x-axis.

Q 19 | Page 84

Find the length of the perpendicular drawn from the origin to the plane 2x − 3y + 6z + 21 = 0.

Q 20 | Page 84

Write the vector equation of the line passing through the point (1, −2, −3) and normal to the plane $\vec{r} \cdot \left( 2 \hat{i} + \hat{j} + 2 \hat{k} \right) = 5 .$

Q 21 | Page 84

Find the vector equation of the plane, passing through the point (abc) and parallel to the plane $\vec{r} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 2$

Q 22 | Page 84

Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is $2 \hat{i} - 3 \hat{j} + 6 \hat{k}$ .

Q 23 | Page 84

Write the equation of a plane which is at a distance of $5\sqrt{3}$ units from origin and the normal to which is equally inclined to coordinate axes.

#### Pages 84 - 86

Q 1 | Page 84

The plane 2x − (1 + λ) y + 3λz = 0 passes through the intersection of the planes

(a) 2x − y = 0 and y − 3z = 0

(b) 2x + 3z = 0 and y = 0

(c) 2x − y + 3z = 0 and y − 3z = 0

(d) None of these

Q 2 | Page 84

The acute angle between the planes 2x − y + z = 6 and x + y + 2z = 3 is

(a) 45°

(b) 60°

(c) 30°

(d) 75°

Q 3 | Page 84

The equation of the plane through the intersection of the planes x + 2y + 3z = 4 and 2x + y − z = −5 and perpendicular to the plane 5x + 3y + 6z + 8 = 0 is

(a) 7x − 2y + 3z + 81 = 0

(b) 23x + 14y − 9z + 48 = 0

(c) 51x − 15y − 50z + 173 = 0

(d) None of these

Q 4 | Page 84

The distance between the planes 2x + 2y − z + 2 = 0 and 4x + 4y − 2z + 5 = 0 is

(a) $\frac{1}{2}$

(b) $\frac{1}{4}$

(c) $\frac{1}{6}$

(d) None of these

Q 5 | Page 84

The image of the point (1, 3, 4) in the plane 2x − y + z + 3 = 0 is

(a) (3, 5, 2)

(b) (−3, 5, 2)

(c) (3, 5, −2)

(d) (3, −5, 2)

Q 6 | Page 85

The equation of the plane containing the two lines

$\frac{x - 1}{2} = \frac{y + 1}{- 1} = \frac{z - 0}{3} and \frac{x}{- 2} = \frac{y - 2}{- 3} = \frac{z + 1}{- 1}$

(a) 8x + y − 5z − 7 = 0

(b) 8x + y + 5z − 7 = 0

(c) 8x − y − 5z − 7 = 0

(d) None of these

Q 7 | Page 85

The equation of the plane $\vec{r} = \hat{i} - \hat{j} + \lambda\left( \hat{i} + \hat{j} + \hat{k} \right) + \mu\left( \hat{i} - 2 \hat{j} + 3 \hat{k} \right)$  in scalar product form is

(a)  $\vec{r} \cdot \left( 5 \hat{i} - 2 \hat{j} - 3 \hat{k} \right) = 7$

(b) $\vec{r} \cdot \left( 5 \hat{i} + 2 \hat{j} - 3 \hat{k} \right) = 7$

(c) $\vec{r} \cdot \left( 5 \hat{i} - 2 \hat{j} + 3 \hat{k} \right) = 7$

(d) None of these

Q 8 | Page 85

The distance of the line $\vec{r} = 2 \hat{i} - 2 \hat{j} + 3 \hat{k} + \lambda\left( \hat{i} - \hat{j}+ 4 \hat{k} \right)$  from the plane $\vec{r} \cdot \left( \hat{i} + 5 \hat{j} + \hat{k} \right) = 5$ is

(a)

$\frac{5}{3\sqrt{3}}$

(b) $\frac{10}{3\sqrt{3}}$

(c) $\frac{25}{3\sqrt{3}}$

(d) None of these

Q 9 | Page 85

The equation of the plane through the line x + y + z + 3 = 0 = 2x − y + 3z + 1 and parallel to the line $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ is

(a) x − 5y + 3z = 7

(b) x − 5y + 3z = −7

(c) x + 5y + 3z = 7

(d) x + 5y + 3z = −7

Q 10 | Page 85

The vector equation of the plane containing the line $\vec{r} = \left( - 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \right) + \lambda\left( 3 \hat{i} - 2 \hat{j} - \hat{k} \right)$ and the point  $\hat{i} + 2 \hat{j} + 3 \hat{k}$  is

(a) $\vec{r} \cdot \left( \hat{i} + 3 \hat{k} \right) = 10$

(b)  $\vec{r} \cdot \left( \hat{i} - 3 \hat{k} \right) = 10$

(c) $\vec{r} \cdot \left( 3\hat{i} - \hat{k} \right) = 10$

(d) None of these

Q 11 | Page 85

A plane meets the coordinate axes at AB and C such that the centroid of ∆ABC is the point (abc). If the equation of the plane is $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = k,$ then k =

(a) 1

(b) 2

(c) 3

(d) None of these

Q 12 | Page 85
The distance between the point (3, 4, 5) and the point where the line $\frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z - 5}{2}$ meets the plane x + y + z = 17 is

(a) 1

(b) 2

(c) 3

(d) None of these

Q 13 | Page 85

A vector parallel to the line of intersection of the planes$\vec{r} \cdot \left( 3 \hat{i} - \hat{j} + \hat{k} \right) = 1 \text{ and } \vec{r} \cdot \left( \hat{i} + 4 \hat{j} - 2 \hat{k} \right) = 2$ is

(a) $- 2 \hat{i} + 7 \hat{j}+ 13 \hat{k}$

(b)  $2 \hat{i} + 7 \hat{j} - 13 \hat{k}$

(c) $-2 \hat{i} + 7 \hat{j} + 13 \hat{k}$

(d) $2 \hat{i} + 7 \hat{j} + 13 \hat{k}$

Q 14 | Page 85

If a plane passes through the point (1, 1, 1) and is perpendicular to the line $\frac{x - 1}{3} = \frac{y - 1}{0} = \frac{z - 1}{4}$ then its perpendicular distance from the origin is

(a) 3/4

(b) 4/3

(c) 7/5

(d) 1

Q 15 | Page 85

The equation of the plane parallel to the lines x − 1 = 2y − 5 = 2z and 3x = 4y − 11 = 3z − 4 and passing through the point (2, 3, 3) is

(a) x − 4y + 2z + 4 = 0

(b) x + 4y + 2z + 4 = 0

(c) x − 4y + 2z − 4 = 0

(d) None of these

Q 16 | Page 86

The distance of the point (−1, −5, −10) from the point of intersection of the line $\vec{r} = 2 \hat{i}- \hat{j} + 2 \hat{k} + \lambda\left( 3 \hat{i} + 4 \hat{j}+ 12 \hat{k} \right)$   and the plane $\vec{r} \cdot \left( \hat{i} - \hat{j} + \hat{k} \right) = 5$ is

(a) 9

(b) 13

(c) 17

(d) None of these

Q 17 | Page 86

The equation of the plane through the intersection of the planes ax + by + cz + d = 0 andlx + my + nz + p = 0 and parallel to the line y=0, z=0

(a) (bl − amy + (cl − anz + dl − ap = 0

(b) (am − blx + (mc − bnz + md − bp = 0

(c) (na − clx + (bn − cmy + nd − cp = 0

(d) None of these

Q 18 | Page 86

The equation of the plane which cuts equal intercepts of unit length on the coordinate axes is

(a) x + y + z = 1

(b) x + y + z = 0

(c) x + y − z = 1

(d) x + y + z = 2

## R.D. Sharma solutions for Class 12 Mathematics chapter 27 - Direction Cosines and Direction Ratios

R.D. Sharma solutions for Class 12 Mathematics chapter 27 (Direction Cosines and Direction Ratios) include all questions with solution and detail explanation from Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session). This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has created the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 27 Direction Cosines and Direction Ratios are Three - Dimensional Geometry Examples and Solutions, Introduction of Three Dimensional Geometry, Equation of a Plane Passing Through Three Non Collinear Points, Relation Between Direction Ratio and Direction Cosines, Intercept Form of the Equation of a Plane, Coplanarity of Two Lines, Distance of a Point from a Plane, Angle Between Line and a Plane, Angle Between Two Planes, Angle Between Two Lines, Vector and Cartesian Equation of a Plane, Shortest Distance Between Two Lines, Equation of a Line in Space, Direction Cosines and Direction Ratios of a Line, Equation of a Plane in Normal Form, Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point, Plane Passing Through the Intersection of Two Given Planes.

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