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.
Will these results work when vectors are represented with column matrices?