Exercise · ZW-ORTHOGONAL-PROJECTION-005 · ★★☆ Intermediate
For \(u=(4,2)\) and \(v=(1,1)\), what is the vector \(u-\operatorname{proj}_v u\)?
Choose one option.
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- C. \((1,-1)\)
Explanation
The projection is \(3(1,1)=(3,3)\), and \(u-(3,3)=(1,-1)\), which is orthogonal to \(v\): \((1,-1)\cdot(1,1)=0\).
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-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)?
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