Exercise · ZW-ORTHOGONAL-PROJECTION-007 · ★★☆ Intermediate
If \(u=c\,v\) with \(c\neq0\), the orthogonal projection of \(u\) onto \(v\) is:
Choose one option.
Show answer
- D. \(u\) itself
Explanation
\(\frac{(cv)\cdot v}{v\cdot v}v=c\,v=u\): a vector on the line of \(v\) is already its own projection.
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-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)?
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