Exercise · ZW-ORTHOGONAL-PROJECTION-008 · ★★☆ Intermediate
Projecting \(u=(5,3)\) onto \(v=(1,1)\) and onto \(2v=(2,2)\) gives:
Choose one option.
Show answer
- B. the same vector in both cases
Explanation
The projection is onto the line spanned, not onto the vector: \(\frac{u\cdot 2v}{2v\cdot2v}\,2v=\frac{u\cdot v}{v\cdot v}\,v\). In both cases it is \((4,4)\).
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-002 Compute the orthogonal projection of u=(2,3) onto v=(1,1).
- 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-009 What is the minimum distance from the point u=(5,0) to the line spanned by v=(1,2)?
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