Contents

These Solutions are part of NCERT Solutions for Class 12 Maths . Here we have given NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry

Board |
CBSE |

Textbook |
NCERT |

Class |
Class 12 |

Subject |
Maths |

Chapter |
Chapter 11 |

Chapter Name |
Three Dimensional Geometry |

Exercise |
Ex 11.1, Ex 11.2, Ex 11.3 |

Number of Questions Solved |
36 |

Category |
NCERT Solutions |

## NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry

### Chapter 11 Three Dimensional Geometry Exercise 11.1

**Question 1.**

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

**Solution:**

Direction angles are 90°, 135°, 45°

Direction cosines are

**Question 2.**

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

**Solution:**

Let direction angle be α each

∴ Direction cosines are cos α, cos α, cos α

But l² + m² + n² = 1

∴cos² a + cos² a + cos² a = 1

**Question 3.**

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

**Solution:**

Now given direction ratios of a line are -8,12,-4

∴ a = -18,b = 12,c = -4

Direction cosines are

**Question 4.**

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

**Solution:**

Let the points be A(2,3,4), B (-1, -2,1), C (5,8,7).

Let direction ratios of AB be

**Question 5.**

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

**Solution:**

The vertices of triangle ABC are A (3, 5, -4), B (-1,1,2), C (-5, -5, -2)

(i) Direction ratios of AB are (-4,-4,6)

Direction cosines are

### Chapter 11 Three Dimensional Geometry Exercise 11.2

**Question 1.**

Show that the three lines with direction cosines:

are mutually perpendicular.

**Solution:**

Let the lines be L1,L2 and L3.

∴ For lines L1 and L2

**Question 2.**

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).

**Solution:**

Let A, B be the points (1, -1, 2), (3, 4, -2) respectively Direction ratios of AB are 2,5, -4

Let C, D be the points (0, 3, 2) and (3, 5, 6) respectively Direction ratios of CD are 3, 2,4 AB is Perpendicular to CD if

**Question 3.**

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

**Solution:**

Let the points be A(4,7,8), B (2,3,4), C (-1,-2,1) andD(1,2,5).

Now direction ratios of AB are

**Question 4.**

Find the equation of the line which passes through the point (1,2,3) and is parallel to the vector

**Solution:**

Equation of the line passing through the point

**Question 5.**

Find the equation of the line in vector and in cartesian form that passes through the point with position vector and is in the direction .

**Solution:**

The vector equation of a line passing through a point with position vector and parallel to the

**Question 6.**

Find the cartesian equation of the line which passes through the point (-2,4, -5) and parallel to the line is given by

**Solution:**

The cartesian equation of the line passing through the point (-2,4, -5) and parallel to the

**Question 7.**

The cartesian equation of a line is

write its vector form.

**Solution:**

The cartesian equation of the line is

Clearly (i) passes through the point (5, – 4, 6) and has 3,7,2 as its direction ratios.

=> Line (i) passes through the point A with

**Question 8.**

Find the vector and the cartesian equations of the lines that passes through the origin and (5,-2,3).

**Solution:**

The line passes through point

Direction ratios of the line passing through the

**Question 9.**

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

**Solution:**

The PQ passes through the point P(3, -2, -5)

**Question 10.**

Find the angle between the following pair of lines

(i)

(ii)

**Solution:**

(i) Let θ be the angle between the given lines.

The given lines are parallel to the vectors

**Question 11.**

Find the angle between the following pair of lines

(i)

(ii)

**Solution:**

Given

(i)

(ii)

**Question 12.**

Find the values of p so that the lines

are at right angles

**Solution:**

The given equation are not in the standard form

The equation of given lines is

**Question 13.**

Show that the lines are perpendicular to each other

**Solution:**

Given lines

…(i)

…(ii)

**Question 14.**

Find the shortest distance between the lines

and

**Solution:**

The shortest distance between the lines

**Question 15.**

Find the shortest distance between the lines

**Solution:**

Shortest distance between the lines

**Question 16.**

Find the distance between die lines whose vector equations are:

and

**Solution:**

Comparing the given equations with

**Question 17.**

Find the shortest distance between the lines whose vector equations are

and

**Solution:**

Comparing these equation with

### Chapter 11 Three Dimensional Geometry Exercise 11.3

**Question 1.**

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

(a) z = 2

(b) x+y+z = 1

(c) 2x + 3y – z = 5

(d) 5y+8 = 0

**Solution:**

(a) Direction ratios of the normal to the plane are 0,0,1

=> a = 0, b = 0, c = 1

**Question 2.**

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector

**Solution:**

**Question 3.**

Find the Cartesian equation of the following planes.

(a)

(b)

(c)

**Solution:**

(a) is the position vector of any arbitrary point P (x, y, z) on the plane.

**Question 4.**

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

(a) 2x + 3y + 4z – 12 = 0

(b) 3y + 4z – 6 = 0

(c) x + y + z = 1

(d) 5y + 8 = 0

**Solution:**

(a) Let N (x1, y1, z1) be the foot of the perpendicular from the origin to the plane 2x+3y+4z-12 = 0

∴ Direction ratios of the normal are 2, 3, 4.

Also the direction ratios of ON are (x1,y1,z1)

**Question 5.**

Find the vector and cartesian equation of the planes

(a) that passes through the point (1,0, -2) and the normal to the plane is

(b) that passes through the point (1,4,6) and the normal vector to the plane is

**Solution:**

(a) Normal to the plane is i + j – k and passes through (1,0,-2)

**Question 6.**

Find the equations of the planes that passes through three points

(a) (1,1,-1) (6,4,-5), (-4, -2,3)

(b) (1,1,0), (1,2,1), (-2,2,-1)

**Solution:**

(a) The plane passes through the points (1,1,-1) (6,4,-5), (-4,-2,3)

Let the equation of the plane passing through(1,1,-1)be

**Question 7.**

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

**Solution:**

Equation of the plane is 2x + y- z = 5 x y z

Dividing by 5:

∴ The intercepts on the axes OX, OY, OZ are , 5, -5 respectively

**Question 8.**

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

**Solution:**

Any plane parallel to ZOX plane is y=b where b is the intercept on y-axis.

∴ b = 3.

Hence equation of the required plane is y = 3.

**Question 9.**

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).

**Solution:**

Given planes are:

3x – y + 2z – 4 = 0 and x + y + z – 2 = 0

Any plane through their intersection is

3x – y + 2z – 4 + λ(x + y + z – 2) = 0

point (2,2,1) lies on it,

∴3 x 2 – 2 + 2 x 1 – 4 +λ(2+2+1-2)=0

=>λ =

Now required equation is 7x – 5y + 4z – 8 = 0

**Question 10.**

Find the vector equation of the plane passing through the intersection of the planes and through the point (2,1,3).

**Solution:**

Equation of the plane passing through the line of intersection of the planes

**Question 11.**

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.

**Solution:**

Given planes are

x + y + z – 1 = 0 …(i)

2x + 3y + 4z – 5 = 0 …(ii)

x – y + z = 0 ….(iii)

**Question 12.**

Find the angle between the planes whose vector equations are

**Solution:**

The angle θ between the given planes is

**Question 13.**

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

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

(b) 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0

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

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

(e) 4x + 8y + z – 8 = 0 and y + z – 4 = 0.

**Solution:**

(a) Direction ratios of the normal of the planes 7x + 5y + 6z + 30 = 0 are 7,5,6

Direction ratios of the normal of the plane 3x – y – 10z + 4 = 0 are 3,-1,-10

The plane 7x + 5y + 6z + 30 = 0 …(i)

3x – y – 10z + y = 0 …(ii)

**Question 14.**

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

Point Plane

(a) (0, 0,0) 3x – 4y + 12z = 3

(b) (3,-2,1) 2x – y + 2z + 3 = 0.

(c) (2,3,-5) x + 2y – 2z = 9

(d) (-6,0,0) 2x – 3y + 6z – 2 = 0

**Solution:**

(a) Given plane: 3x – 4y + 12z – 3 = 0

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