Inner product spaces
Lesson 9 of 10
Orthogonal projection
Objective. Project one vector onto another.
The key idea
proj_u(v) = ((v · u)/(u · u)) u. The leftover v − proj_u(v) is orthogonal to u.
Worked example
Project v = (3, 4) onto u = (1, 0).
- 1v · u = 3 and u · u = 1.
- 2Scalar is 3.
- 33 × (1, 0) = (3, 0).
Answer: (3, 0)
Practice
Each question comes with a picture and a listen button. Try it first, then reveal the answer.
Common mistake. Dividing by |u| once rather than by u · u.
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