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# RD Sharma solutions for Class 12 Maths chapter 28 - Straight Line in Space [Latest edition]

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## Chapter 28: Straight Line in Space

Ex. 28.1Ex. 28.2Ex. 28.3Ex. 28.4Ex. 28.5Very Short AnswersMCQ

### RD Sharma solutions for Class 12 Maths Chapter 28 Straight Line in Space 28.1 [Pages 9 - 10]

Ex. 28.1 | 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  $3 \hat{i} + 2 \hat{j} - 8 \hat{k} .$

Ex. 28.1 | Q 2 | Page 9

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

Ex. 28.1 | 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.

Ex. 28.1 | 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.

Ex. 28.1 | 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.

Ex. 28.1 | 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).

Ex. 28.1 | 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.

Ex. 28.1 | Q 8 | Page 10

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

Ex. 28.1 | 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.

Ex. 28.1 | 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.

Ex. 28.1 | 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.

Ex. 28.1 | 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.

Ex. 28.1 | 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.

Ex. 28.1 | 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).

Ex. 28.1 | 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.

Ex. 28.1 | 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  $\frac{- x - 2}{1} = \frac{y + 3}{7} = \frac{2z - 6}{3} .$

Ex. 28.1 | 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.

Ex. 28.1 | 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.

### RD Sharma solutions for Class 12 Maths Chapter 28 Straight Line in Space 28.2 [Pages 15 - 17]

Ex. 28.2 | 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.

Ex. 28.2 | 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).

Ex. 28.2 | 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).

Ex. 28.2 | 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  $\frac{x + 3}{3} = \frac{y - 4}{5} = \frac{z + 8}{6} .$

Ex. 28.2 | Q 5 | Page 16

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.

Ex. 28.2 | 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).

Ex. 28.2 | Q 7 | Page 16

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

Ex. 28.2 | Q 8.1 | Page 16

Find the angle between the following pair of line:

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

Ex. 28.2 | Q 8.2 | Page 16

Find the angle between the following pair of line:

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

Ex. 28.2 | Q 8.3 | Page 16

Find the angle between the following pair of line:

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

Ex. 28.2 | Q 9.1 | Page 16

Find the angle between the following pair of line:

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

Ex. 28.2 | Q 9.2 | Page 16

Find the angle between the following pair of line:

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

Ex. 28.2 | Q 9.3 | Page 16

Find the angle between the following pair of line:

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

Ex. 28.2 | Q 9.4 | Page 16

Find the angle between the following pair of line:

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

Ex. 28.2 | Q 9.5 | Page 16

Find the angle between the following pair of line:

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

Ex. 28.2 | Q 9.6 | Page 16

Find the angle between the following pair of line:

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

Ex. 28.2 | Q 10.1 | Page 16

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

Ex. 28.2 | 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 .

Ex. 28.2 | 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 .

Ex. 28.2 | Q 10.4 | Page 16

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

Ex. 28.2 | 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).

Ex. 28.2 | Q 12 | Page 16

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

Ex. 28.2 | Q 13 | Page 16

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

Ex. 28.2 | Q 14 | Page 16

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

Ex. 28.2 | Q 15 | Page 16

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

Ex. 28.2 | 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  $\overrightarrow{r} = \hat{i} + \lambda\left( 2 \hat{i} + \hat{j} - 3 \hat{k} \right) \text { and } \overrightarrow{r} = \left( 2 \hat{i} + \hat{j} - \hat{ k} \right) + \mu\left( \hat{i} + \hat{j} + \hat{k} \right) .$

Ex. 28.2 | 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).

Ex. 28.2 | Q 18 | Page 17

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

Ex. 28.2 | 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.

Ex. 28.2 | 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.

Ex. 28.2 | Q 21 | Page 17

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

Ex. 28.2 | 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

Ex. 28.2 | Q 23 | Page 17

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

Ex. 28.2 | 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.

### RD Sharma solutions for Class 12 Maths Chapter 28 Straight Line in Space 28.3 [Pages 22 - 23]

Ex. 28.3 | Q 1 | Page 22

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

Ex. 28.3 | Q 2 | Page 22

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

Ex. 28.3 | Q 3 | Page 22

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

Ex. 28.3 | 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.

Ex. 28.3 | 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.

Ex. 28.3 | Q 6.1 | Page 22

Determine whether the following pair of lines intersect or not:

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

Ex. 28.3 | Q 6.2 | Page 22

Determine whether the following pair of lines intersect or not:

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

Ex. 28.3 | Q 6.3 | Page 22

Determine whether the following pair of lines intersect or not:

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

Ex. 28.3 | Q 6.4 | Page 22

Determine whether the following pair of lines intersect or not:

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

Ex. 28.3 | 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.

### RD Sharma solutions for Class 12 Maths Chapter 28 Straight Line in Space 28.4 [Pages 29 - 30]

Ex. 28.4 | Q 1 | Page 29

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

Ex. 28.4 | 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.

Ex. 28.4 | 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).

Ex. 28.4 | 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

Ex. 28.4 | 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.

Ex. 28.4 | 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

Ex. 28.4 | Q 7 | Page 30

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

Ex. 28.4 | 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  $\overrightarrow{r} = \hat{j} + 2 \hat{k} + \lambda\left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) .$  Also, find the length of the perpendicular

Ex. 28.4 | Q 9 | Page 30

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

Ex. 28.4 | Q 10 | Page 30

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

Ex. 28.4 | Q 11 | Page 30

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

Ex. 28.4 | 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

Ex. 28.4 | Q 13 | Page 30

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

Ex. 28.4 | 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).

### RD Sharma solutions for Class 12 Maths Chapter 28 Straight Line in Space 28.5 [Pages 37 - 38]

Ex. 28.5 | Q 1.1 | Page 37

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

Ex. 28.5 | Q 1.2 | Page 37

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

Ex. 28.5 | Q 1.3 | Page 37

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

Ex. 28.5 | Q 1.4 | Page 37

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

Ex. 28.5 | Q 1.5 | Page 37

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

Ex. 28.5 | Q 1.6 | Page 37

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

Ex. 28.5 | Q 1.7 | Page 37

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

Ex. 28.5 | Q 1.8 | Page 37

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

Ex. 28.5 | Q 2.1 | Page 38

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

Ex. 28.5 | Q 2.2 | Page 38

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

Ex. 28.5 | Q 2.3 | Page 38

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

Ex. 28.5 | Q 2.4 | Page 38

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

Ex. 28.5 | Q 3.1 | Page 38

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

Ex. 28.5 | Q 3.2 | Page 38

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

Ex. 28.5 | Q 3.3 | Page 38

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

Ex. 28.5 | Q 3.4 | Page 38

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

Ex. 28.5 | Q 4.1 | Page 38

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

Ex. 28.5 | Q 4.2 | Page 38

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

Ex. 28.5 | 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)

Ex. 28.5 | 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)

Ex. 28.5 | Q 6 | Page 38

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

Ex. 28.5 | Q 7.1 | Page 38

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

Ex. 28.5 | Q 7.2 | Page 38

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

Ex. 28.5 | Q 7.3 | Page 38

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

Ex. 28.5 | Q 7.4 | Page 38

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

Ex. 28.5 | Q 8 | Page 38

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

### RD Sharma solutions for Class 12 Maths Chapter 28 Straight Line in Space Very Short Answers [Pages 41 - 42]

Very Short Answers | Q 1 | Page 41

Write the cartesian and vector equations of X-axis.

Very Short Answers | Q 2 | Page 41

Write the cartesian and vector equations of Y-axis.

Very Short Answers | Q 3 | Page 41

Write the cartesian and vector equations of Z-axis.

Very Short Answers | Q 4 | Page 41

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

Very Short Answers | 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.

Very Short Answers | Q 6 | Page 41

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

Very Short Answers | Q 7 | Page 41

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

Very Short Answers | 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.

Very Short Answers | Q 9 | Page 41

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

Very Short Answers | Q 10 | Page 41

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

Very Short Answers | Q 11 | Page 41

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

Very Short Answers | Q 12 | Page 41

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

Very Short Answers | Q 13 | Page 41

Write the formula for the shortest distance between the lines

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

Very Short Answers | Q 14 | Page 41

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

Very Short Answers | 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

Very Short Answers | 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

Very Short Answers | Q 17 | Page 41

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

Very Short Answers | 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.

Very Short Answers | 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 $\frac{x + 3}{3} = \frac{4 - y}{5} = \frac{z + 8}{6} .$

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

Very Short Answers | Q 21 | Page 42

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

### RD Sharma solutions for Class 12 Maths Chapter 28 Straight Line in Space MCQ [Pages 42 - 43]

MCQ | Q 1 | Page 42

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

• a) 45°

• (b) 30°

• (c) 60°

• (d) 90°

MCQ | 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

MCQ | Q 3 | Page 42

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

• (a) 4, 5, 7

• (b) 4, −5, 7

• (c) 4, −5, −7

• (d) −4, 5, 7

MCQ | Q 4 | Page 42

The angle between the lines

$\frac{x - 1}{1} = \frac{y - 1}{1} = \frac{z - 1}{2} \text{ and }, \frac{x - 1}{- \sqrt{3} - 1} = \frac{y - 1}{\sqrt{3} - 1} = \frac{z - 1}{4}$ is
• (a) $\cos^{- 1} \left( \frac{1}{65} \right)$

• (b) $\frac{\pi}{6}$

• (c) $\frac{\pi}{3}$

• (d) $\frac{\pi}{4}$

MCQ | Q 5 | Page 43

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

• 3, 1, −2

• 2, −4, 1

•  $\frac{3}{\sqrt{14}}, \frac{1}{\sqrt{14}}, \frac{- 2}{\sqrt{14}}$

•  $\frac{2}{\sqrt{41}}, \frac{- 4}{\sqrt{41}}, \frac{1}{\sqrt{41}}$

MCQ | 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

• 7

• 5

• 0

•  none of these

MCQ | 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

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

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

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

•  none of these

MCQ | 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 =

•  −2

•  −1

•  1

•  2

MCQ | Q 9 | Page 43

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

•  $\frac{1}{\sqrt{14}}, - \frac{3}{\sqrt{14}}, \frac{2}{\sqrt{14}}$

•  $\frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}}$

•  $- \frac{1}{\sqrt{14}}, \frac{3}{\sqrt{14}}, \frac{2}{\sqrt{14}}$

•  $- \frac{1}{\sqrt{14}}, - \frac{2}{\sqrt{14}}, - \frac{3}{\sqrt{14}}$

MCQ | Q 10 | Page 43

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

• π/2

• π/3

•  π/4

•  5π/12

MCQ | 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

• $13; \frac{12}{13}, \frac{4}{13}, \frac{3}{13}$

• $19; \frac{12}{19}, \frac{4}{19}, \frac{3}{19}$

• $11; \frac{12}{11}, \frac{14}{11}, \frac{3}{11}$

•  none of these

MCQ | Q 12 | Page 43

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

• parallel

• intersecting

•  skew

•  coincident

MCQ | Q 13 | Page 43

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

•  parallel to x-axis

•  parallel to y-axis

•  parallel to z-axis

•  perpendicular to z-axis

MCQ | Q 14 | Page 43

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

•  $\sqrt{30}$

• $2\sqrt{30}$

• $5\sqrt{30}$

•  $3\sqrt{30}$

## Chapter 28: Straight Line in Space

Ex. 28.1Ex. 28.2Ex. 28.3Ex. 28.4Ex. 28.5Very Short AnswersMCQ

## RD Sharma solutions for Class 12 Maths chapter 28 - Straight Line in Space

RD Sharma solutions for Class 12 Maths chapter 28 (Straight Line in Space) include all questions with solution and detail explanation. 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 the CBSE Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Maths chapter 28 Straight Line in Space 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, 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, Shortest Distance Between Two Lines, Equation of a Line in Space, Direction Cosines and Direction Ratios of a Line.

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