Question
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
Solution
The equations of given planes are
4x + 8 y + z – 8 = 0 ...(1)
and 0x + y + z – 4 = 0 ...(2)
Direction ratios of normal to plane (1) are 4, 8, 1.
Direction ratios of normal to plane (2) are 0, 1, 1.
Now, 
∴ given planes (1) and (2) are not parallel.
Again (4) (0) + (8) (1) + (1) (1) = 0 + 8 + 1 = 9 ≠ 0
∴ given planes (1) and (2) are not perpendicular to each other.
Let θ be angle between planes.
