Will the results of this chapter work when vectors are represented with column matrices?

A good answer might be:

Thankfully, yes.


Example

Time for an example. In the diagram the answer can simply be read off the graph paper. But pretend you didn't notice that.

The vector w is represented by by (6, 5)T. The vector v is represented by by (9, 0)T. Find kv and u.

  1. Compute the lengths:
    | w |  =   ((6, 5)T·(6, 5)T)   =   7.81
    | v |  =   ((9, 0)T·(9, 0)T)   =   9
  2. Compute the unit vectors:
    wu  =   (6, 5)T / 7.81
    vu  =   (9, 0)T / 9  =   (1, 0)T
  3. Compute the cosine of the angle between the vectors:
    wu · vu  =   (1/7.81) (6, 5)T · (1, 0)T  =   6/7.81
  4. Assemble the projection:
    kv  =   | w | (wu·vu) vu
    kv  =   7.81 (6/7.81) (1, 0)T     =   6(1, 0)T  =   (6, 0)T
  5. Compute the orthogonal vector:
    u  =   w - kv
    u  =   (6, 5)T - (6, 0)T  =   (0, 5)T

The result is that u  =   (6, 0)T + (0, 5)T, as expected. Of course, the example was easy.

QUESTION 8:

Was it actually necessary to compute | w |?