Question

Suppose plane 1 is defined by the Cartesian equation 5x-y-3=0 and the plane
2 is defined by the equation x+ky-3z=0. Are there any values of k that will make the planes parallel If so, what values? If not, explain why

Answer 1

The planes are parallel if their normals are scalar multiples of each other, that is:

<5,-1,0> = c<1,k,-3>

then 5 = c
-1 = kc
0 = -3c
from the last , c = 0, but that contradicts our c = 5 condition.

So there is no value of k that will make the 2 planes parallel.