Example

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

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

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:

   w_u  =   (6, 5)^T / 7.81

   v_u  =   (9, 0)^T / 9  =   (1, 0)^T

3. Compute the cosine of the angle between the vectors:

   w_u · v_u  =   (1/7.81) (6, 5)^T · (1, 0)^T  =   6/7.81

4. Assemble the projection:

   kv  =   | w | (w_u·v_u) v_u

   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 w  =   (6, 0)^T + (0, 5)^T, as expected. Of course, the example was easy.