w projected onto v is NOT the same as v projected onto w
The vector resulting from a projection is oriented in the direction of the vector projected onto.
After you have found kv it is easy to find the orthogonal vector, u:
w = kv + u, so u = w - kv
In the above formula, "+" means vector addition and "-" means vector subtraction.
So far in this chapter the vectors have all been geometrical vectors. There has been no mention of coordinate frames or column matrices.