Inner product spaces
Lesson 9 of 10
Standard practice
Orthogonal projection
Objective. Project one vector onto another.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: 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)
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Dividing by |u| once rather than by u · u.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.