# NCERT Solutions for Class 12 Maths Chapter 11- Three Dimensional Geometry Exercise 11.3

## NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.3 â€“ CBSE Term II Free PDF Download

The Exercise 11.3 of NCERT Solutions for Class 12 Maths Chapter 11- Three Dimensional Geometry is based on the following topics:

1. Plane
1. Equation of a plane in normal form
2. Equation of a plane perpendicular to a given vector, passing through given point
3. Equation of a plane passing through three non collinear points
4. Intercept form of the equation of a plane
5. Plane passing through the intersection of two given planes
2. Coplanarity of Two Lines
3. Angle between Two Planes
4. Distance of a Point from a Plane
5. Angle between a Line and a Plane

Solving the problems of this exercise will help the students in understanding the problem solving methods related to the topics mentioned above.

## Download PDF of NCERT Solutions for Class 12 Maths Chapter 11- Three Dimensional Geometry Exercise 11.3

### Access Other Exercises of Class 12 Maths Chapter 11

Exercise 11.1 Solutions 5 Questions

Exercise 11.2 Solutions 17 Questions

Miscellaneous Exercise On Chapter 11 Solutions 23 Questions

#### Access Answers to NCERT Class 12 Maths Chapter 11 Exercise 11.3

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 â€“ z = 5

(d) 5y + 8 = 0

Solution:

(a) z = 2

Given:

The equation of the plane, z = 2 or 0x + 0y + z = 2 â€¦. (1)

Direction ratio of the normal (0, 0, 1)

By using the formula,

âˆš[(0)2 + (0)2 + (1)2] = âˆš1

= 1

Now,

Divide both the sides of equation (1) by 1, we get

0x/(1) + 0y/(1) + z/1 = 2

So this is of the form lx + my + nz = d

Where, l, m, n are the direction cosines and d is the distance

âˆ´ The direction cosines are 0, 0, 1

Distance (d) from the origin is 2 units

(b) x + y + z = 1

Given:

The equation of the plane, x + y + z = 1â€¦. (1)

Direction ratio of the normal (1, 1, 1)

By using the formula,

âˆš[(1)2 + (1)2 + (1)2] = âˆš3

Now,

Divide both the sides of equation (1) by âˆš3, we get

x/(âˆš3) + y/(âˆš3) + z/(âˆš3) = 1/âˆš3

So this is of the form lx + my + nz = d

Where, l, m, n are the direction cosines and d is the distance

âˆ´ The direction cosines are 1/âˆš3, 1/âˆš3, 1/âˆš3

Distance (d) from the origin is 1/âˆš3 units

(c) 2x + 3y â€“ z = 5

Given:

The equation of the plane, 2x + 3y â€“ z = 5â€¦. (1)

Direction ratio of the normal (2, 3, -1)

By using the formula,

âˆš[(2)2 + (3)2 + (-1)2] = âˆš14

Now,

Divide both the sides of equation (1) by âˆš14, we get

2x/(âˆš14) + 3y/(âˆš14) â€“ z/(âˆš14) = 5/âˆš14

So this is of the form lx + my + nz = d

Where, l, m, n are the direction cosines and d is the distance

âˆ´ The direction cosines are 2/âˆš14, 3/âˆš14, -1/âˆš14

Distance (d) from the origin is 5/âˆš14 units

(d) 5y + 8 = 0

Given:

The equation of the plane, 5y + 8 = 0

-5y = 8 or

0x â€“ 5y + 0z = 8â€¦. (1)

Direction ratio of the normal (0, -5, 0)

By using the formula,

âˆš[(0)2 + (-5)2 + (0)2] = âˆš25

= 5

Now,

Divide both the sides of equation (1) by 5, we get

0x/(5) â€“ 5y/(5) â€“ 0z/(5) = 8/5

So this is of the form lx + my + nz = d

Where, l, m, n are the direction cosines and d is the distance

âˆ´ The direction cosines are 0, -1, 0

Distance (d) from the origin is 8/5 units

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:

3. Find the Cartesian equation of the following planes:
(a)

Solution:

Given:

The equation of the plane.

4. In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
(a) 2x + 3y + 4z â€“ 12 = 0

(b) 3y + 4z â€“ 6 = 0

(c) x + y + z = 1

(d) 5y + 8 = 0

Solution:

(a) 2x + 3y + 4z â€“ 12 = 0

Let the coordinate of the foot ofÂ âŠ¥Â P from the origin to the given plane be P(x, y, z).

2x + 3y + 4z = 12 â€¦. (1)

Direction ratio are (2, 3, 4)

âˆš[(2)2 + (3)2 + (4)2] = âˆš(4 + 9 + 16)

= âˆš29

Now,

Divide both the sides of equation (1) by âˆš29, we get

2x/(âˆš29) + 3y/(âˆš29) + 4z/(âˆš29) = 12/âˆš29

So this is of the form lx + my + nz = d

Where, l, m, n are the direction cosines and d is the distance

âˆ´ The direction cosines are 2/âˆš29, 3/âˆš29, 4/âˆš29

Coordinate of the foot (ld, md, nd) =

= [(2/âˆš29) (12/âˆš29), (3/âˆš29) (12/âˆš29), (4/âˆš29) (12/âˆš29)]

= 24/29, 36/29, 48/29

(b) 3y + 4z â€“ 6 = 0

Let the coordinate of the foot ofÂ âŠ¥Â P from the origin to the given plane be P(x, y, z).

0x + 3y + 4z = 6 â€¦. (1)

Direction ratio are (0, 3, 4)

âˆš[(0)2 + (3)2 + (4)2] = âˆš(0 + 9 + 16)

= âˆš25

= 5

Now,

Divide both the sides of equation (1) by 5, we get

0x/(5) + 3y/(5) + 4z/(5) = 6/5

So this is of the form lx + my + nz = d

Where, l, m, n are the direction cosines and d is the distance

âˆ´ The direction cosines are 0/5, 3/5, 4/5

Coordinate of the foot (ld, md, nd) =

= [(0/5) (6/5), (3/5) (6/5), (4/5) (6/5)]

= 0, 18/25, 24/25

(c) x + y + z = 1

Let the coordinate of the foot ofÂ âŠ¥Â P from the origin to the given plane be P(x, y, z).

x + y + z = 1 â€¦. (1)

Direction ratio are (1, 1, 1)

âˆš[(1)2 + (1)2 + (1)2] = âˆš(1 + 1 + 1)

= âˆš3

Now,

Divide both the sides of equation (1) by âˆš3, we get

1x/(âˆš3) + 1y/(âˆš3) + 1z/(âˆš3) = 1/âˆš3

So this is of the form lx + my + nz = d

Where, l, m, n are the direction cosines and d is the distance

âˆ´ The direction cosines are 1/âˆš3, 1/âˆš3, 1/âˆš3

Coordinate of the foot (ld, md, nd) =

= [(1/âˆš3) (1/âˆš3), (1/âˆš3) (1/âˆš3), (1/âˆš3) (1/âˆš3)]

= 1/3, 1/3, 1/3

(d) 5y + 8 = 0

Let the coordinate of the foot ofÂ âŠ¥Â P from the origin to the given plane be P(x, y, z).

0x â€“ 5y + 0z = 8 â€¦. (1)

Direction ratio are (0, -5, 0)

âˆš[(0)2 + (-5)2 + (0)2] = âˆš(0 + 25 + 0)

= âˆš25

= 5

Now,

Divide both the sides of equation (1) by 5, we get

0x/(5) â€“ 5y/(5) + 0z/(5) = 8/5

So this is of the form lx + my + nz = d

Where, l, m, n are the direction cosines and d is the distance

âˆ´ The direction cosines are 0, -1, 0

Coordinate of the foot (ld, md, nd) =

= [(0/5) (8/5), (-5/5) (8/5), (0/5) (8/5)]

= 0, -8/5, 0

5. Find the vector and Cartesian equations of the planes

(a) that passes through the point (1, 0, â€“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

Â

Solution:

x â€“ 1 â€“ 2y + 8 + z â€“ 6 = 0

x â€“ 2y + z + 1 = 0

x â€“ 2y + z = -1

âˆ´ The required Cartesian equation of the plane is x â€“ 2y + z = -1

x â€“ 1 â€“ 2y + 8 + z â€“ 6 = 0

x â€“ 2y + z + 1 = 0

x â€“ 2y + z = -1

âˆ´ The required Cartesian equation of the plane is x â€“ 2y + z = -1

6. Find the equations of the planes that passes through three points.
(a) (1, 1, â€“1), (6, 4, â€“5), (â€“4, â€“2, 3)

(b) (1, 1, 0), (1, 2, 1), (â€“2, 2, â€“1)

Solution:

Given:

The points are (1, 1, -1), (6, 4, -5), (-4, -2, 3).

Let,

= 1(12 â€“ 10) â€“ 1(18 â€“ 20) -1 (-12 + 16)

= 2 + 2 â€“ 4

= 0

Since, the value of determinant is 0.

âˆ´ The points are collinear as there will be infinite planes passing through the given 3 points.

(b) (1, 1, 0), (1, 2, 1), (â€“2, 2, â€“1)

7. Find the intercepts cut off by the plane 2x + y â€“ z = 5.

Solution:

Given:

The plane 2x + y â€“ z = 5

Let us express the equation of the plane in intercept form

x/a + y/b + z/c = 1

Where a, b, c are the intercepts cut-off by the plane at x, y and z axes respectively.

2x + yÂ â€“Â z = 5 â€¦. (1)

Now divide both the sides of equation (1) by 5, we get

2x/5 + y/5 â€“ z/5 = 5/5

2x/5 + y/5 â€“ z/5 = 1

x/(5/2) + y/5 + z/(-5) = 1

Here, a = 5/2, b = 5 and c = -5

âˆ´ The intercepts cut-off by the plane are 5/2, 5 and -5.

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

Solution:

We know that the equation of the plane ZOX is y = 0

So, the equation of plane parallel to ZOX is of the form, y = a

Since the y-intercept of the plane is 3, a = 3

âˆ´ The required equation of the plane is y = 3

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

Solution:

Given:

Equation of the plane passes through the intersection of the plane is given by

(3x â€“ y + 2z â€“ 4) +Â Î» (x + y + z â€“ 2) = 0 and the plane passes through the points (2, 2, 1).

So, (3 Ã— 2 â€“ 2 + 2 Ã— 1 â€“ 4) +Â Î» (2 + 2 + 1Â â€“Â 2) = 0

2 + 3Î»Â = 0

3Î»Â = -2

Î» = -2/3 â€¦. (1)

Upon simplification, the required equation of the plane is given as

(3x â€“ y + 2z â€“ 4) â€“ 2/3 (x + y + z â€“ 2) = 0

(9x â€“ 3y + 6z â€“ 12 â€“ 2x â€“ 2y â€“ 2z + 4)/3 = 0

7x â€“ 5y + 4z â€“ 8 = 0

âˆ´ The required equation of the plane is 7x â€“ 5y + 4z â€“ 8 = 0

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

Solution:

The equation of any plane through the intersection of the planes given in equations (1) and (2) is given by,

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 â€“ y + z = 0.

Solution:

Let the equation of the plane that passes through the two-given planes

x + y + z = 1 and 2x + 3y + 4z = 5 is

(x + y + z â€“ 1) +Â Î» (2x + 3y + 4zÂ â€“Â 5) = 0

(2Î»Â + 1) x + (3Î»Â + 1) y + (4Î»Â + 1) z -1 â€“ 5Î»Â = 0â€¦â€¦ (1)

So the direction ratio of the plane is (2Î»Â + 1, 3Î»Â + 1, 4Î»Â + 1)

And direction ratio of another plane is (1, -1, 1)

Since, both the planes areÂ âŠ¥

So by substituting in a1a2 + b1b2 + c1c2 = 0

(2Î»Â + 1Â Ã—Â 1) + (3Î»Â + 1Â Ã—Â (-1)) + (4Î»Â + 1 Ã— 1) = 0

2Î»Â + 1 â€“ 3Î»Â â€“ 1 + 4Î»Â + 1 = 0

3Î» + 1 = 0

Î» = -1/3

Substitute the value ofÂ Î»Â in equation (1) we get,

x â€“ z + 2 = 0

âˆ´ The required equation of the plane is x â€“ z + 2 = 0

12. Find the angle between the planes whose vector equations are

Solution:

13. In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
(a) 7x + 5y + 6z + 30 = 0 and 3x â€“ y â€“ 10z + 4 = 0

(b) 2x + y + 3z â€“ 2 = 0 and x â€“ 2y + 5 = 0

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

(d) 2x â€“ 2y + 4z + 5 = 0 and 3x â€“ 3y + 6z â€“ 1 = 0

(e) 4x + 8y + z â€“ 8 = 0 and y + z â€“ 4 = 0

Solution:

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

Given:

The equation of the given planes are

7x + 5y + 6z + 30 = 0 and 3x â€“ y â€“ 10z + 4 = 0

Two planes areÂ âŠ¥Â if the direction ratio of the normal to the plane is

a1a2Â + b1b2Â + c1c2Â = 0

21 â€“ 5 â€“ 60

-44 â‰  0

Both the planes are notÂ âŠ¥Â to each other.

Now, two planes are || to each other if the direction ratio of the normal to the plane is

âˆ´ The angle is cos-1 (2/5)

(b) 2x + y + 3z â€“ 2 = 0 and x â€“ 2y + 5 = 0

Given:

The equation of the given planes are

2x + y + 3z â€“ 2 = 0 and x â€“ 2y + 5 = 0

Two planes areÂ âŠ¥Â if the direction ratio of the normal to the plane is

a1a2Â + b1b2Â + c1c2Â = 0

2 Ã— 1 + 1 Ã— (-2) + 3 Ã— 0

= 0

âˆ´ The given planes areÂ âŠ¥Â to each other.

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

Given:

The equation of the given planes are

2x â€“ 2y + 4z + 5 =0 and x â€“ 2y + 5 = 0

We know that, two planes areÂ âŠ¥Â if the direction ratio of the normal to the plane is

a1a2Â + b1b2Â + c1c2Â = 0

6 + 6 + 24

36 â‰  0

âˆ´Â Both the planes are notÂ âŠ¥Â to each other.

Now let us check, both planes are || to each other if the direction ratio of the normal to the plane is

âˆ´ The given planes are || to each other.

(d) 2x â€“ 2y + 4z + 5 = 0 and 3x â€“ 3y + 6z â€“ 1 = 0

Given:

The equation of the given planes are

2x â€“ y + 3z â€“ 1 = 0 and 2x â€“ y + 3z + 3 = 0

We know that, two planes areÂ âŠ¥Â if the direction ratio of the normal to the plane is

a1a2Â + b1b2Â + c1c2Â = 0

2 Ã— 2 + (-1) Ã— (-1) + 3 Ã— 3

14 â‰  0

âˆ´Â Both the planes are notÂ âŠ¥Â to each other.

Now, let us check two planes are || to each other if the direction ratio of the normal to the plane is

âˆ´ The given planes are || to each other.

(e) 4x + 8y + z â€“ 8 = 0 and y + z â€“ 4 = 0

Given:

The equation of the given planes are

4x + 8y + z â€“ 8 = 0 and y + z â€“ 4 = 0

We know that, two planes areÂ âŠ¥Â if the direction ratio of the normal to the plane is

a1a2Â + b1b2Â + c1c2Â = 0

0 + 8 + 1

9 â‰  0

âˆ´Â Both the planes are notÂ âŠ¥Â to each other.

Now let us check, two planes are || to each other if the direction ratio of the normal to the plane is

âˆ´Â Both the planes are not || to each other.

Now let us find the angle between them which is given as

âˆ´ The angle is 45o.

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 â€“ 4y + 12 z = 3

(b) (3, -2, 1) 2x â€“ y + 2z + 3 = 0

(c) (2, 3, -5) x + 2y â€“ 2z = 9

(d) (-6, 0, 0) 2x â€“ 3y + 6z â€“ 2 = 0

Solution:

(a) Point Plane

(0, 0, 0) 3x â€“ 4y + 12 z = 3

We know that, distance of point P(x1, y1, z1) from the plane Ax + By + Cz â€“ D = 0 is given as:

Given point is (0, 0, 0) and the plane is 3x â€“ 4y + 12z = 3

= |3/âˆš169|

= 3/13

âˆ´ The distance is 3/13.

(b) Point Plane

(3, -2, 1) 2x â€“ y + 2z + 3 = 0

We know that, distance of point P(x1, y1, z1) from the plane Ax + By + Cz â€“ D = 0 is given as:

Given point is (3, -2, 1) and the plane is 2x â€“ y + 2z + 3 = 0

= |13/âˆš9|

= 13/3

âˆ´ The distance is 13/3.

(c) Point Plane

(2, 3, -5) x + 2y â€“ 2z = 9

We know that, distance of point P(x1, y1, z1) from the plane Ax + By + Cz â€“ D = 0 is given as:

Given point is (2, 3, -5) and the plane is x + 2y â€“ 2z = 9

= |9/âˆš9|

= 9/3

= 3

âˆ´ The distance is 3.

(d) Point Plane

(-6, 0, 0) 2x â€“ 3y + 6z â€“ 2 = 0

We know that, distance of point P(x1, y1, z1) from the plane Ax + By + Cz â€“ D = 0 is given as:

Given point is (-6, 0, 0) and the plane is 2x â€“ 3y + 6z â€“ 2 = 0

= |14/âˆš49|

= 14/7

= 2

âˆ´ The distance is 2.