Math Help - Projecting a Vector Onto a Plane

1. Projecting a Vector Onto a Plane

In a linear algebra course, I am learning a technique for finding the the projection of a vector onto a plane (3D space) without using cross-products or dot products. The "direction of projection" can be, but does not have to be normal to the plane. The idea is to find a 3x3 matrix that will transform a source vector into its projection vector.

I have two exercises. I did the first without difficultly, but the second stumps me. I keep getting an answer that is not a vector in the plane. Where did I go wrong?

See the attached pdf file for details.