# NCERT solutions for Mathematics Exemplar Class 12 chapter 11 - Three Dimensional Geometry [Latest edition]

## Chapter 11: Three Dimensional Geometry

Solved ExamplesExercise
Solved Examples [Pages 224 - 234]

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 11 Three Dimensional GeometrySolved Examples [Pages 224 - 234]

Solved Examples | Q 1 | Page 224

If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.

Solved Examples | Q 2 | Page 224

Find the direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4).

Solved Examples | Q 3 | Page 225

If a line makes an angle of 30°, 60°, 90° with the positive direction of x, y, z-axes, respectively, then find its direction cosines.

Solved Examples | Q 4 | Page 225

The x-coordinate of a point on the line joining the points Q(2, 2, 1) and R(5, 1, –2) is 4. Find its z-coordinate.

Solved Examples | Q 5 | Page 225

Find the distance of the point whose position vector is (2hat"i" + hat"j" - hat"k") from the plane vec"r" * (hat"i" - 2hat"j" + 4hat"k") = 9

Solved Examples | Q 6 | Page 226

Find the distance of the point (– 2, 4, – 5) from the line (x + 3)/3 = (y - 4)/5 = (z + 8)/6

Solved Examples | Q 7 | Page 226

Find the coordinates of the point where the line through (3, – 4, – 5) and (2, –3, 1) crosses the plane passing through three points (2, 2, 1), (3, 0, 1) and (4, –1, 0)

Solved Examples | Q 8 | Page 227

Find the distance of the point (–1, –5, – 10) from the point of intersection of the line vec"r" = 2hat"i" - hat"j" + 2hat"k" + lambda(3hat"i" + 4hat"j" + 2hat"k") and the plane vec"r" * (hat"i" - hat"j" + hat"k") = 5

Solved Examples | Q 9 | Page 227

A plane meets the co-ordinates axis in A, B, C such that the centroid of the ∆ABC is the point (α, β, γ). Show that the equation of the plane is x/alpha + y/beta + z/ϒ = 3

Solved Examples | Q 10 | Page 228

Find the angle between the lines whose direction cosines are given by the equations: 3l + m + 5n = 0 and 6mn – 2nl + 5lm = 0.

Solved Examples | Q 11 | Page 229

Find the co-ordinates 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).

Solved Examples | Q 12 | Page 230

Find the image of the point (1, 6, 3) in the line x/1 = (y - 1)/2 = (z - 2)/3.

Solved Examples | Q 13 | Page 231

Find the image of the point having position vector hat"i" + 3hat"j" + 4hat"k" in the plane hat"r" * (2hat"i" - hat"j" + hat"k") + 3 = 0.

#### Objective Type Questions from 14 to 19

Solved Examples | Q 14 | Page 232

The coordinates of the foot of the perpendicular drawn from the point (2, 5, 7) on the x-axis are given by ______.

• (2, 0, 0)

• (0, 5, 0)

• (0, 0, 7)

• (0, 5, 7)

Solved Examples | Q 15 | Page 232

P is a point on the line segment joining the points (3, 2, –1) and (6, 2, –2). If x co-ordinate of P is 5, then its y co-ordinate is ______.

• 2

• 1

• –1

• –2

Solved Examples | Q 16 | Page 232

If α, β, γ are the angles that a line makes with the positive direction of x, y, z axis, respectively, then the direction cosines of the line are ______.

• sin α, sin β, sin γ

• cos α, cos β, cos γ

• tan α, tan β, tan γ

• cos2α, cos2β, cos2γ

Solved Examples | Q 17 | Page 233

The distance of a point P(a, b, c) from x-axis is ______.

• sqrt("a"^2 + "c"^2)

• sqrt("a"^2 + "b"^2)

• sqrt("b"^2 + "c"^2)

• b2 + c2

Solved Examples | Q 18 | Page 233

The equations of x-axis in space are ______.

• x = 0, y = 0

• x = 0, z = 0

• x = 0

• y = 0, z = 0

Solved Examples | Q 19 | Page 233

A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.

• +-(1, 1, 1)

• +-(1/sqrt(3), 1/sqrt(3), 1/sqrt(3))

• +-(1/3, 1/3, 1/3)

• +-(1/sqrt(3), (-1)/sqrt(3), (-1)/sqrt(3))

#### Fill in the blanks from 20 to 22

Solved Examples | Q 20 | Page 233

If a line makes angles pi/2, 3/4 pi and pi/4 with x, y, z axis, respectively, then its direction cosines are ______.

Solved Examples | Q 21 | Page 234

If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin2α + sin2β + sin2γ is ______.

Solved Examples | Q 22 | Page 234

If a line makes an angle of pi/4 with each of y and z-axis, then the angle which it makes with x-axis is ______.

#### State whether the following statement is True or False: 23 to 24

Solved Examples | Q 23 | Page 234

The points (1, 2, 3), (–2, 3, 4) and (7, 0, 1) are collinear.

• True

• False

Solved Examples | Q 24 | Page 234

The vector equation of the line passing through the points (3, 5, 4) and (5, 8, 11) is vec"r" = 3hat"i" + 5hat"j" + 4hat"k" + lambda(2hat"i" + 3hat"j" + 7hat"k")

• True

• False

Exercise [Pages 235 - 240]

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 11 Three Dimensional GeometryExercise [Pages 235 - 240]

Exercise | Q 1 | Page 235

Find the position vector of a point A in space such that vec"OA" is inclined at 60º to OX and at 45° to OY and |vec"OA"| = 10 units.

Exercise | Q 2 | Page 235

Find the vector equation of the line which is parallel to the vector 3hat"i" - 2hat"j" + 6hat"k" and which passes through the point (1, –2, 3).

Exercise | Q 3 | Page 235

Show that the lines (x - 1)/2 = (y - 2)/3 = (z - 3)/4 and (x - 4)/5 = (y - 1)/2 = z intersect. Also, find their point of intersection.

Exercise | Q 4 | Page 235

Find the angle between the lines vec"r" = 3hat"i" - 2hat"j" + 6hat"k" + lambda(2hat"i" + hat"j" + 2hat"k") and vec"r" = (2hat"j" - 5hat"k") + mu(6hat"i" + 3hat"j" + 2hat"k")

Exercise | Q 5 | Page 235

Prove that the line through A(0, – 1, – 1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(– 4, 4, 4).

Exercise | Q 6 | Page 235

Prove that the lines x = py + q, z = ry + s and x = p′y + q′, z = r′y + s′ are perpendicular if pp′ + rr′ + 1 = 0.

Exercise | Q 7 | Page 235

Find the equation of a plane which bisects perpendicularly the line joining the points A(2, 3, 4) and B(4, 5, 8) at right angles.

Exercise | Q 8 | Page 235

Find the equation of a plane which is at a distance 3sqrt(3) units from origin and the normal to which is equally inclined to coordinate axis.

Exercise | Q 9 | Page 235

If the line drawn from the point (–2, – 1, – 3) meets a plane at right angle at the point (1, – 3, 3), find the equation of the plane.

Exercise | Q 10 | Page 235

Find the equation of the plane through the points (2, 1, 0), (3, –2, –2) and (3, 1, 7).

Exercise | Q 11 | Page 236

Find the equations of the two lines through the origin which intersect the line (x - 3)/2 = (y - 3)/1 = z/1 at angles of pi/3 each.

Exercise | Q 12 | Page 236

Find the angle between the lines whose direction cosines are given by the equations l + m + n = 0, l2 + m2 – n2 = 0.

Exercise | Q 13 | Page 236

If a variable line in two adjacent positions has direction cosines l, m, n and l + δl, m + δm, n + δn, show that the small angle δθ between the two positions is given by δθ2 = δl2 + δm2 + δn

Exercise | Q 14 | Page 236

O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.

Exercise | Q 15 | Page 236

Two systems of rectangular axis have the same origin. If a plane cuts them at distances a, b, c and a′, b′, c′, respectively, from the origin, prove that 1/"a"^2 + 1/"b"^2 + 1/"c"^2 = 1/"a'"^2 + 1/"b'"^2 + 1/"c'"^2

Exercise | Q 16 | Page 236

Find the foot of perpendicular from the point (2, 3, –8) to the line (4 - x)/2 = y/6 = (1 - z)/3. Also, find the perpendicular distance from the given point to the line.

Exercise | Q 17 | Page 236

Find the distance of a point (2, 4, –1) from the line (x + 5)/1 = (y + 3)/4 = (z - 6)/(-9)

Exercise | Q 18 | Page 236

Find the length and the foot of perpendicular from the point (1, 3/2, 2) to the plane 2x – 2y + 4z + 5 = 0.

Exercise | Q 19 | Page 236

Find the equations of the line passing through the point (3, 0, 1) and parallel to the planes x + 2y = 0 and 3y – z = 0.

Exercise | Q 20 | Page 237

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

Exercise | Q 21 | Page 237

Find the shortest distance between the lines given by vec"r" = (8 + 3lambdahat"i" - (9 + 16lambda)hat"j" + (10 + 7lambda)hat"k" and vec"r" = 15hat"i" + 29hat"j" + 5hat"k" + mu(3hat"i" + 8hat"j" - 5hat"k")

Exercise | Q 22 | Page 237

Find the equation of the plane which 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 and 2x + y – z + 5 = 0.

Exercise | Q 23 | Page 237

The plane ax + by = 0 is rotated about its line of intersection with the plane z = 0 through an angle α. Prove that the equation of the plane in its new position is "a"x + "b"y +- (sqrt("a"^2 + "b"^2) tan alpha)z = 0.

Exercise | Q 24 | Page 237

Find the equation of the plane through the intersection of the planes vec"r" * (hat"i" + 3hat"j") - 6 = 0 and vec"r" * (3hat"i" - hat"j" - 4hat"k") = 0, whose perpendicular distance from origin is unity.

Exercise | Q 25 | Page 237

Show that the points (hat"i" - hat"j" + 3hat"k") and 3(hat"i" + hat"j" + hat"k") are equidistant from the plane vec"r" * (5hat"i" + 2hat"j" - 7hat"k") + 9 = 0 and lies on opposite side of it.

Exercise | Q 26 | Page 237

vec"AB" = 3hat"i" - hat"j" + hat"k" and vec"CD" = -3hat"i" + 2hat"j" + 4hat"k" are two vectors. The position vectors of the points A and C are 6hat"i" + 7hat"j" + 4hat"k" and -9hat"j" + 2hat"k", respectively. Find the position vector of a point P on the line AB and a point Q on the line Cd such that vec"PQ" is perpendicular to vec"AB" and vec"CD" both.

Exercise | Q 27 | Page 237

Show that the straight lines whose direction cosines are given by 2l + 2m – n = 0 and mn + nl + lm = 0 are at right angles.

Exercise | Q 28 | Page 237

If l1, m1, n1; l2, m2, n2; l3, m3, n3 are the direction cosines of three mutually perpendicular lines, prove that the line whose direction cosines are proportional to l1 + l2 + l3, m1 + m2 + m3, n1 + n2 + n3 makes equal angles with them.

#### Objective Type Questions from 29 to 36

Exercise | Q 29 | Page 237

Distance of the point (α, β, γ) from y-axis is ____________.

• β

• |β|

• |b| + |γ|

• sqrt("a"^2 + γ^2)

Exercise | Q 30 | Page 238

If the directions cosines of a line are k,k,k, then ______.

• k > 0

• 0 < k < 1

• k = 1

• k = 1/sqrt(3) or - 1/sqrt(3)

Exercise | Q 31 | Page 238

The distance of the plane vec"r" *(2/7hat"i" + 3/4hat"j" - 6/7hat"k") = 1 from the origin is ______.

• 1

• 7

• 1/7

• None of these

Exercise | Q 32 | Page 238

The sine of the angle between the straight line (x - 2)/3 = (y - 3)/4 = (z - 4)/5 and the plane 2x – 2y + z = 5 is ______.

• 10/(6sqrt(5))

• 4/(5sqrt(2))

• (2sqrt(3))/5

• sqrt(2)/10

Exercise | Q 33 | Page 238

The reflection of the point (α, β, γ) in the xy-plane is ______.

• (α, β, 0)

• (0, 0, γ)

• (–α, –β, γ)

• (α, β, –γ)

Exercise | Q 34 | Page 238

The area of the quadrilateral ABCD, where A(0,4,1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2), is equal to ______.

• 9 sq.units

• 18 sq.units

• 27 sq.units

• 81 sq.units

Exercise | Q 35 | Page 238

The locus represented by xy + yz = 0 is ______.

• A pair of perpendicular lines

• A pair of parallel lines

• A pair of parallel planes

• A pair of perpendicular planes

Exercise | Q 36 | Page 238

The plane 2x – 3y + 6z – 11 = 0 makes an angle sin–1(α) with x-axis. The value of α is equal to ______.

• sqrt(3)/2

• sqrt(2)/3

• 2/7

• 3/7

#### Fill in the blanks 37 to 41

Exercise | Q 37 | Page 239

A plane passes through the points (2, 0, 0) (0, 3, 0) and (0, 0, 4). The equation of plane is ______.

Exercise | Q 38 | Page 239

The direction cosines of vector (2hat"i" + 2hat"j" - hat"k") are ______.

Exercise | Q 39 | Page 239

The vector equation of the line (x - 5)/3 = (y + 4)/7 = (z - 6)/2 is ______.

Exercise | Q 40 | Page 239

The vector equation of the line through the points (3, 4, –7) and (1, –1, 6) is ______.

Exercise | Q 41 | Page 239

The Cartesian equation of the plane vec"r" * (hat"i" + hat"j" - hat"k") = 2 is ______.

#### State whether the following statement is True or False: 42 to 49

Exercise | Q 42 | Page 239

The unit vector normal to the plane x + 2y +3z – 6 = 0 is 1/sqrt(14)hat"i" + 2/sqrt(14)hat"j" + 3/sqrt(14)hat"k".

• True

• False

Exercise | Q 43 | Page 239

The intercepts made by the plane 2x – 3y + 5z + 4 = 0 on the coordinate axes are -2, 4/3, (-4)/5.

• True

• False

Exercise | Q 44 | Page 239

The angle between the line vec"r" = (5hat"i" - hat"j" - 4hat"k") + lambda(2hat"i" - hat"j" + hat"k") and the plane vec"r".(3hat"i" - 4hat"j" - hat"k") + 5 = 0 is sin^-1 (5/(2sqrt(91))).

• True

• False

Exercise | Q 45 | Page 239

The angle between the planes vec"r".(2hat"i" - 3hat"j" + hat"k") = 1 and vec"r"(hat"i" - hat"j") = 4 is cos^-1 ((-5)/sqrt(58)).

• True

• False

Exercise | Q 46 | Page 239

The line vec"r" = 2hat"i" - 3hat"j" - hat"k" + lambda(hat"i" - hat"j" + 2hat"k") lies in the plane vec"r".(3hat"i" + hat"j" - hat"k") + 2 = 0.

• True

• False

Exercise | Q 47 | Page 239

The vector equation of the line (x - 5)/3 = (y + 4)/7 = (z - 6)/2 is vec"r" = 5hat"i" - 4hat"j" + 6hat"k" + lambda(3hat"i" + 7hat"j" + 2hat"k").

• True

• False

Exercise | Q 48 | Page 240

The equation of a line, which is parallel to 2hat"i" + hat"j" + 3hat"k" and which passes through the point (5, –2, 4), is (x - 5)/2 = (y + 2)/(-1) = (z - 4)/3.

• True

• False

Exercise | Q 49 | Page 240

If the foot of perpendicular drawn from the origin to a plane is (5, – 3, – 2), then the equation of plane is vec"r".(5hat"i" - 3hat"j" - 2hat"k") = 38.

• True

• False

## Chapter 11: Three Dimensional Geometry

Solved ExamplesExercise

## NCERT solutions for Mathematics Exemplar Class 12 chapter 11 - Three Dimensional Geometry

NCERT solutions for Mathematics Exemplar Class 12 chapter 11 (Three Dimensional Geometry) 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 Mathematics Exemplar Class 12 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics Exemplar Class 12 chapter 11 Three Dimensional Geometry 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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