Question

Find all unit vectors parallel to the yz-plane that are perpendicular to the vector (3, -1, 2)

Answer 1

if the vector is parallel to the yz plane, it must have 0 for an x component. Making an educated guess on a perpendicular vector, it would be (0, 2, 1). taking the inner product (or dot product if you like physics) of these 2 vectors gives 0, so they are perpendicular.

Answer 2

however, (0, 2, 1) is not a unit vector. Its length is:\[\sqrt{0^2+2^2+1^2}=\sqrt{5}\]
so divide all the entries of the vector by root 5 and you have a unit vector.