Another Example
You don't have to compute the length of w.
It cancels out before the final answer.
And, all you need is the square of the length of v, |v|^{2}.
Some books show formulae for projection that make use of these
facts (but, to my taste, are less intuitive).
Here is another example, this time not so easy.
The vector w is represented by by (3.2, 7)^{T}.
The vector v is represented by by (8, 4)^{T}.
Find kv and u.
- Compute the lengths:
- | w | = (keep it symbolic)
- | v |^{2} =
(8, 4)^{T}·(8, 4)^{T}
= 80
- Compute the unit vectors:
- w_{u} = (3.2, 7)^{T} / | w |
- v_{u} = (8, 4)^{T} / | v |
- Compute the cosine of the angle between the vectors:
- w_{u} · v_{u}
= (3.2, 7)^{T} / | w | · (8, 4)^{T} / | v |
= 53.6/( | w || v |)
- Assemble the projection:
- kv = | w | (w_{u}·v_{u}) v_{u}
- kv = | w | [53.6 / (| w || v |)] (8, 4)^{T} / | v |
- kv = 53.6 / (| v |) (8, 4)^{T} / | v |
- kv = 53.6 / (| v |^{2}) (8, 4)^{T}
- kv = 53.6 / 80 (8, 4)^{T}
- kv = ((53.6*8)/80, (53.6*4)/80)^{T}
= ( 5.3, 2.68)^{T}
- Compute the orthogonal vector:
- u = w - kv
- u = (3.2, 7)^{T} - (5.3, 2.68)^{T}
= (-2.1, 4.32)^{T}