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

#### Chapters ## Solutions for Chapter 11: Three Dimensional Geometry

Below listed, you can find solutions for Chapter 11 of CBSE, Karnataka Board PUC NCERT for Class 12 Maths.

Exercise 11.1Exercise 11.2Exercise 11.3Exercise 11.4
Exercise 11.1 [Pages 467 - 478]

### NCERT solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Exercise 11.1 [Pages 467 - 478]

Exercise 11.1 | Q 1 | Page 467

If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.

Exercise 11.1 | Q 2 | Page 467

Find the direction cosines of a line which makes equal angles with the coordinate axes.

Exercise 11.1 | Q 3 | Page 467

If a line has the direction ratios −18, 12, −4, then what are its direction cosines?

Exercise 11.1 | Q 4 | Page 477

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector.3hati+2hatj-2hatk

Exercise 11.1 | Q 4 | Page 467

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

Exercise 11.1 | Q 5 | Page 477

Find the equation of the line in vector and in Cartesian form that passes through the point with position vector 2hati -hatj+4hatk  and is in the direction hati + 2hatj - hatk.

Exercise 11.1 | Q 5 | Page 467

Find the Direction Cosines of the Sides of the Triangle Whose Vertices Are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2)

Exercise 11.1 | Q 6 | Page 477

Find the Cartesian equation of the line which passes through the point (−2, 4, −5) and parallel to the line given by (x+3)/3 = (y-4)/5 = ("z"+8)/6

Exercise 11.1 | Q 7 | Page 477

The Cartesian equation of a line is (x-5)/3 = (y+4)/7 = ("z"-6)/2 Write its vector form.

The given line passes through the point (5, −4, 6). The position vector of this point is veca = 5hati - 4hatj + 6hatk

Also, the direction ratios of the given line are 3, 7, and 2.

This means that the line is in the direction of vector, vecb =3hati +7hatj + 2hatk

It is known that the line through position vector veca and in the direction of the vector vecbis given by the equation, vecr = veca+lambdavecb, lambda in R This is the required equation of the given line in vector form.

Exercise 11.1 | Q 8 | Page 477

Find the vector and the Cartesian equations of the lines that pass through the origin and (5, −2, 3).

Exercise 11.1 | Q 9 | Page 478

Find the vector and the Cartesian equations of the line that passes through the points (3, −2, −5), (3, −2, 6).

Exercise 11.2 [Pages 477 - 478]

### NCERT solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Exercise 11.2 [Pages 477 - 478]

Exercise 11.2 | Q 1 | Page 477

Show that the three lines with direction cosines 12/13,(-3)/13,(-4)/13; 4/13,12/13,3/13;3/13,(-4)/13,12/13  are mutually perpendicular.

Exercise 11.2 | Q 2 | Page 477

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

Exercise 11.2 | Q 3 | Page 477

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

Exercise 11.2 | Q 10.1 | Page 478

Find the angle between the following pairs of lines:

vecr = 2hati - 5hatj + hatk + lambda(3hati - 2hatj + 6hatk) and vecr = 7hati - 6hatk + mu(hati + 2hatj + 2hatk)

Exercise 11.2 | Q 10.2 | Page 478

Find the angle between the following pairs of lines:

vecr = 3hati + hatj - 2hatk + lambda(hati - hatj - 2hatk) and vecr = 2hati - hatj -56hatk + mu(3hati - 5hatj - 4hatk)

Exercise 11.2 | Q 11.1 | Page 478

Find the angle between the following pairs of lines:

(x-2)/2 = (y-1)/5 = (z+3)/(-3) and (x+2)/(-1) = (y-4)/8 = (z -5)/4

Exercise 11.2 | Q 11.2 | Page 478

Find the angle between the following pairs of lines:

x/y = y/2 = z/1 and (x-5)/4 = (y-2)/1 = (z - 3)/8

Exercise 11.2 | Q 12 | Page 478

Find the values of p so the line (1-x)/3 = (7y-14)/2p = (z-3)/2 and (7-7x)/(3p) = (y -5)/1 = (6-z)/5 are at right angles.

Exercise 11.2 | Q 13 | Page 478

Show that the lines (x-5)/7 = (y + 2)/(-5) = z/1 and x/1 = y/2 = z/3 are perpendicular to each other.

Exercise 11.2 | Q 14 | Page 478

Find the shortest distance between the lines

vecr = (hati+2hatj+hatk) + lambda(hati-hatj+hatk) and vecr = 2hati - hatj - hatk + mu(2hati + hatj + 2hatk)

Exercise 11.2 | Q 15 | Page 478

Find the shortest distance between the lines (x+1)/7 = (y+1)/(-6) = (z+1)/1 and (x-3)/1 = (y-5)/(-2) = (z-7)/1

Exercise 11.2 | Q 16 | Page 478

Find the shortest distance between the lines whose vector equations are vecr = (hati + 2hatj + 3hatk) + lambda(hati - 3hatj + 2hatk) and vecr = 4hati + 5hatj + 6hatk + mu(2hati + 3hatj + hatk)

Exercise 11.2 | Q 17 | Page 478

Find the shortest distance between the lines whose vector equations are

vecr = (1-t)hati + (t - 2)hatj + (3 -2t)hatk and vecr = (s+1)hati + (2s + 1)hatk

Exercise 11.3 [Pages 493 - 494]

### NCERT solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Exercise 11.3 [Pages 493 - 494]

Exercise 11.3 | Q 1 | Page 493

In following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

z = 2

Exercise 11.3 | Q 1.2 | Page 493

In following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

x + y + z = 1

Exercise 11.3 | Q 1.3 | Page 493

In following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

2x + 3y – z = 5

Exercise 11.3 | Q 1.4 | Page 493

In following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

5y + 8 = 0

Exercise 11.3 | Q 2 | Page 493

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector.3hati + 5hatj - 6hatk

Exercise 11.3 | Q 3 | Page 493

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

x + y + z = 1

Exercise 11.3 | Q 3.1 | Page 493

Find the Cartesian equation of the following planes:

vecr.(hati + hatj-hatk) = 2

Exercise 11.3 | Q 3.2 | Page 493

Find the Cartesian equation of the following planes:

vecr.(2hati + 3hatj-4hatk) = 1

Exercise 11.3 | Q 3.3 | Page 493

Find the Cartesian equation of the following planes:

vecr.[(s-2t)hati + (3 - t)hatj + (2s + t)hatk] = 15

Exercise 11.3 | Q 4.1 | Page 493

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

2x + 3y + 4z – 12 = 0

Exercise 11.3 | Q 4.2 | Page 493

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

3y + 4z – 6 = 0

Exercise 11.3 | Q 4.4 | Page 493

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

5y + 8 = 0

Exercise 11.3 | Q 5.1 | Page 493

Find the vector and Cartesian equation of the planes that passes through the point (1, 0, −2) and the normal to the plane is hati + hatj - hatk

Exercise 11.3 | Q 5.2 | Page 493

Find the vector and Cartesian equation of the planes that passes through the point (1, 4, 6) and the normal vector to the plane is hati -2hatj +  hatk

Exercise 11.3 | Q 6.1 | Page 493

Find the equations of the planes that passes through three points.

(1, 1, −1), (6, 4, −5), (−4, −2, 3)

Exercise 11.3 | Q 6.2 | Page 493

Find the equations of the planes that passes through three points.

(1, 1, 0), (1, 2, 1), (−2, 2, −1)

Exercise 11.3 | Q 7 | Page 493

Find the intercepts cut off by the plane 2x + y – z = 5.

Exercise 11.3 | Q 8 | Page 493

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

Exercise 11.3 | Q 9 | Page 493

Find the equation of the plane through the intersection of the planes 3x – y + 2z – 4 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).

Exercise 11.3 | Q 10 | Page 493

Find the vector equation of the plane passing through the intersection of the planes vecr.(2hati + 2hatj - 3hatk) = 7, vecr.(2hati + 5hatj + 3hatk) = 9 and through the point (2, 1, 3)

Exercise 11.3 | Q 11 | Page 493

Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x − y + z = 0. Also find the distance of the plane, obtained above, from the origin.

Exercise 11.3 | Q 12 | Page 494

Find the angle between the planes whose vector equations are vecr.(2hati + 2hatj - 3hatk) = 5 and hatr.(3hati - 3hatj  + 5hatk) = 3

Exercise 11.3 | Q 13.1 | Page 494

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0

Exercise 11.3 | Q 13.2 | Page 494

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

2x + y + 3z – 2 = 0 and x – 2y + 5 = 0

Exercise 11.3 | Q 13.3 | Page 494

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0

Exercise 11.3 | Q 13.4 | Page 494

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

2x – y + 3z – 1 = 0 and 2x – y + 3z + 3 = 0

Exercise 11.3 | Q 13.5 | Page 494

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

4x + 8y + z – 8 = 0 and y + z – 4 = 0

Exercise 11.3 | Q 14.1 | Page 494

In the given cases, find the distance of each of the given points from the corresponding given plane.

Point                    Plane
(0, 0, 0)           3x – 4y + 12 z = 3

Exercise 11.3 | Q 14.2 | Page 494

In the given cases, find the distance of each of the given points from the corresponding given plane

Point                   Plane

(3, – 2, 1)             2x – y + 2z + 3 = 0

Exercise 11.3 | Q 14.3 | Page 494

In the given cases, find the distance of each of the given points from the corresponding given plane.

Point                 Plane

(2, 3, – 5)           x + 2y – 2z = 9

Exercise 11.3 | Q 14.4 | Page 494

In the given cases, find the distance of each of the given points from the corresponding given plane.

Point              Plane

(– 6, 0, 0)        2x – 3y + 6z – 2 = 0

Exercise 11.4 [Pages 497 - 499]

### NCERT solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Exercise 11.4 [Pages 497 - 499]

Exercise 11.4 | Q 1 | Page 497

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), (4, 3, – 1).

Exercise 11.4 | Q 2 | Page 497

If l1m1n1 and l2m2n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m1n2 − m2n1n1l2 − n2l1l1m2 ­− l2m1.

Exercise 11.4 | Q 3 | Page 498

Find the angle between the lines whose direction ratios are aband b − cc − aa − b.

Exercise 11.4 | Q 4 | Page 498

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

Exercise 11.4 | Q 5 | Page 498

If the coordinates of the points A, B, C, D 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.

Exercise 11.4 | Q 6 | Page 498

If the lines (x-1)/(-3) = (y -2)/(2k) = (z-3)/2 and (x-1)/(3k) = (y-1)/1 = (z -6)/(-5) are perpendicular, find the value of k.

Exercise 11.4 | Q 7 | Page 498

Find the vector equation of the plane passing through (1, 2, 3) and perpendicular to the plane vecr.(hati + 2hatj -5hatk) + 9 = 0

Exercise 11.4 | Q 8 | Page 498

Find the equation of the plane passing through (abc) and parallel to the plane vecr.(hati + hatj + hatk) = 2

Exercise 11.4 | Q 9 | Page 498

Find the shortest distance between lines vecr = 6hati + 2hatj + 2hatk + lambda(hati - 2hatj + 2hatk) and vecr =-4hati - hatk + mu(3hati - 2hatj - 2hatk)

Exercise 11.4 | Q 10 | Page 498

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

Exercise 11.4 | Q 11 | Page 498

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

Exercise 11.4 | Q 12 | Page 498

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

Exercise 11.4 | Q 13 | Page 498

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

Exercise 11.4 | Q 14 | Page 498

If the points (1, 1, p) and (−3, 0, 1) be equidistant from the plane vecr.(3hati + 4hatj - 12hatk)+ 13 = 0, then find the value of p.

Exercise 11.4 | Q 15 | Page 498

Find the equation of the plane passing through the line of intersection of the planes vecr.(hati + hatj + hatk) = 1 and vecr.(2hati + 3hatj -hatk) + 4 = 0 and parallel to x-axis.

Exercise 11.4 | Q 16 | Page 498

If O be the origin and the coordinates of P be (1, 2, −3), then find the equation of the plane passing through P and perpendicular to OP.

Exercise 11.4 | Q 17 | Page 498

Find the equation of the plane which contains the line of intersection of the planes vecrr.(hati + 2hatj + 3hatk) - 4 = 0, vecr.(2hati + htj - hatk) + 5 = 0,  and which is perpendicular to the plane vecr.(5hati + 3hatj - 6hatk) + 8 = 0.

Exercise 11.4 | Q 18 | Page 499

Find the distance of the point (−1, −5, −­10) from the point of intersection of the line vecr = 2hati -hatj + 2hatk + lambda(3hati + 4hatj + 2hatk) and the plane vecr.(hati -hatj + hatk) = 5.

Exercise 11.4 | Q 19 | Page 499

Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes vecr = (hati - hatj + 2hatk)  = 5and vecr.(3hati + hatj + hatk) = 6

Exercise 11.4 | Q 20 | Page 499

Find the vector equation of the line passing through the point (1, 2, − 4) and perpendicular to the two lines:  (x -8)/3 = (y+19)/(-16) = (z - 10)/7 and (x - 15)/3 = (y - 29)/8 = (z- 5)/(-5)

Exercise 11.4 | Q 21 | Page 499

Prove that if a plane has the intercepts abc and is at a distance of P units from the origin, then 1/a^2 + 1/b^2 + 1/c^2 = 1/p^2

Exercise 11.4 | Q 22 | Page 499

Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is

(A) 2 units

(B) 4 units

(C) 8 units

(D)2/sqrt29 "units"

Exercise 11.4 | Q 23 | Page 499

The planes: 2− y + 4z = 5 and 5x − 2.5y + 10z = 6 are

(A) Perpendicular

(B) Parallel

(C) intersect y-axis

(C) passes through (0,0,5/4)

## Solutions for Chapter 11: Three Dimensional Geometry

Exercise 11.1Exercise 11.2Exercise 11.3Exercise 11.4 ## NCERT solutions for Class 12 Maths chapter 11 - Three Dimensional Geometry

Shaalaa.com has the CBSE, Karnataka Board PUC Mathematics Class 12 Maths CBSE, Karnataka Board PUC solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. NCERT solutions for Mathematics Class 12 Maths CBSE, Karnataka Board PUC 11 (Three Dimensional Geometry) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.

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Concepts covered in Class 12 Maths 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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