Question -
Answer -
(a) 7x + 5y + 6z + 30 = 0and 3x – y – 10z + 4 = 0
Given:
The equation of thegiven planes are
7x + 5y + 6z + 30 = 0and 3x – y – 10z + 4 = 0
Two planes are ⊥ if the direction ratio of the normal tothe plane is
a1a2 +b1b2 + c1c2 = 0
21 – 5 – 60
-44 ≠ 0
Both the planes arenot ⊥ to each other.
Now, two planes are ||to each other if the direction ratio of the normal to the plane is
∴ The angle is cos-1 (2/5)
(b) 2x + y + 3z – 2 = 0and x – 2y + 5 = 0
Given:
The equation of thegiven planes are
2x + y + 3z – 2 = 0and x – 2y + 5 = 0
Two planes are ⊥ if the direction ratio of the normal tothe plane is
a1a2 +b1b2 + c1c2 = 0
2 × 1 + 1 × (-2) + 3 ×0
= 0
∴ The given planesare ⊥ to each other.
(c) 2x – 2y + 4z + 5 = 0and 3x – 3y + 6z – 1 = 0
Given:
The equation of thegiven planes are
2x – 2y + 4z + 5 =0and x – 2y + 5 = 0
We know that, twoplanes are ⊥ if the directionratio of the normal to the plane is
a1a2 +b1b2 + c1c2 = 0
6 + 6 + 24
36 ≠ 0
∴ Both the planesare not ⊥ to each other.
Now let us check, bothplanes are || to each other if the direction ratio of the normal to the planeis
∴ The given planes are|| to each other.
(d) 2x – 2y + 4z + 5 = 0and 3x – 3y + 6z – 1 = 0
Given:
The equation of thegiven planes are
2x – y + 3z – 1 = 0and 2x – y + 3z + 3 = 0
We know that, twoplanes are ⊥ if the directionratio of the normal to the plane is
a1a2 +b1b2 + c1c2 = 0
2 × 2 + (-1) × (-1) +3 × 3
14 ≠ 0
∴ Both the planesare not ⊥ to each other.
Now, let us check twoplanes are || to each other if the direction ratio of the normal to the planeis
∴ The given planes are|| to each other.
(e) 4x + 8y + z – 8 = 0and y + z – 4 = 0
Given:
The equation of thegiven planes are
4x + 8y + z – 8 = 0and y + z – 4 = 0
We know that, twoplanes are ⊥ if the directionratio of the normal to the plane is
a1a2 +b1b2 + c1c2 = 0
0 + 8 + 1
9 ≠ 0
∴ Both the planesare not ⊥ to each other.
Now let us check, twoplanes are || to each other if the direction ratio of the normal to the planeis
∴ Both the planesare not || to each other.
Now let us find theangle between them which is given as
∴ The angle is 45o.