Calculus Planes

Question

Jamila is designing a computer game where two planes need to be aligned so they are perpendicular to each other. Help Jamila determine and value of k such that the planes 2x-ky+2kz-4=0 and 3x+y-z=2 are perpendicular.

Answer 1

If two planes are perpendicular, then their normal vectors must also be perpendicular. First we find the normal vectors of the planes, which is the gradient of the function.

plane 1: n₁ = [2, -k, 2k]. plane 2: n₂ = [3, 1, -1].

If two vectors are to be perpendicular, then they must have a dot product of 0. Set this equation up and solve for k

n₁ • n₂ = [2, -k, 2k] • [3, 1, -1] = 6 -k - 2k = 6 - 3k = 0 ==> k = 2