Exercise · ZW-ORTHOGONAL-PROJECTION-002 · ★☆☆ Basic
Compute the orthogonal projection of \(u=(2,3)\) onto \(v=(1,1)\).
Choose one option.
Show answer
- A. \(\left(\tfrac52,\tfrac52\right)\)
Explanation
\(u\cdot v=5\) and \(v\cdot v=2\), so the projection is \(\tfrac52(1,1)\). Forgetting to divide by \(v\cdot v\) gives \((5,5)\).
Review the concept: Orthogonal projection
The orthogonal projection of a vector onto a nonzero vector is the multiple of the latter closest to the former; the remainder is perpendicular to the latter.
\[\operatorname{proj}_{v}u=\frac{u\cdot v}{v\cdot v}\,v\]
More practice problems: orthogonal projection
- ZW-ORTHOGONAL-PROJECTION-001 Compute the orthogonal projection of u=(3,4) onto v=(1,0).
- ZW-ORTHOGONAL-PROJECTION-003 Compute the orthogonal projection of u=(1,2,3) onto v=(2,0,1).
- ZW-ORTHOGONAL-PROJECTION-004 What is the length of the orthogonal projection of u=(1,2,2) onto v=(2,1,2)?
- ZW-ORTHOGONAL-PROJECTION-005 For u=(4,2) and v=(1,1), what is the vector u-proj_v u?
- ZW-ORTHOGONAL-PROJECTION-006 If v≠0, the orthogonal projection of u onto v is the zero vector exactly when:
- ZW-ORTHOGONAL-PROJECTION-007 If u=c v with c≠0, the orthogonal projection of u onto v is:
- ZW-ORTHOGONAL-PROJECTION-008 Projecting u=(5,3) onto v=(1,1) and onto 2v=(2,2) gives:
- ZW-ORTHOGONAL-PROJECTION-009 What is the minimum distance from the point u=(5,0) to the line spanned by v=(1,2)?
← ZW-ORTHOGONAL-PROJECTION-001 ZW-ORTHOGONAL-PROJECTION-003 →