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:
2x + y + 3z – 2 = 0 and x – 2y + 5 = 0
Solution
The equations of given planes are
2x + y + 3z – 2 = 0 ...(1)
and x – 2y + 5 = 0 ...(2)
Direction ratios of normal to plane (1) are 2, 1, 3
Direction ratios of normal to plane (2) are 1, – 2, 0.
Now, 
∴ planes (1) and (2) are not parallel.
Again (2) (1) + (1) (– 2) + (3) (0) = 2 – 2 + 0 = 0
∴ given planes (1) and (2) are perpendicular to each other.
