## Rajasthan Board RBSE Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.7

Question 1.

Find the angle between the planes :

Solution:

Question 2.

Find the angle between the planes :

(i) x + y + 2z = 9 and 2x – y + z = 15

(ii) 2 x – y + z = 4 and x + y + 2z = 3

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

Solution:

If plane are a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 then angle between them is

Question 3.

Show that following planes are at right angles :

Solution:

x – 2y + 4z = 10 and 18x + 17y + 4z = 49

a_{1} = 1,b_{1} = -2, c_{1} = 4 and a_{2} = 18 b_{2} = 17, c_{2} = 4

(i) Planes will be perpendicular if

a_{1} a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0

L.H.S. = 1 × 18 + (-2) × 17 + 4 × 4

= 18 – 34+16

= – 34 + 34

= 0

∴ L.H.S. = R.H.S.

Question 4.

Find λ, if following planes are perpendicular to each other:

Solution:

Question 5.

Find the angle between the line \(\frac { x+1 }{ 3 } \) = \(\frac { y-1 }{ 2 } \) = \(\frac { z-2 }{ 4 } \) and the plane 2x + y – 3z + 4 = 0

Solution:

Normal vector of the plane 2x + y – 3z + 4 = 0

Question 6.

Find the ngle between the line \(\frac { x-2 }{ 3 } \) = \(\frac { y+1 }{ -1 } \) = \(\frac { z-3 }{ 2 } \) and the plane 3x + 4y + z + 5 = 0.

Solution:

Question 7.

Find the angle between

Solution:

We know that the angle between the line and the plane is

Question 8.

Find the angle between

Solution:

We know that angle between the line

Question 9.

Determine the value of m, if line

Solution:

Question 10.

Find m, if line

is parallel to the plane

Solution:

If given line is parallel to given plane