Exercise · ZW-ORTHOGONAL-PROJECTION-009 · ★★★ Advanced
What is the minimum distance from the point \(u=(5,0)\) to the line spanned by \(v=(1,2)\)?
Choose one option.
Show answer
- C. \(2\sqrt5\)
Explanation
The closest point is the projection \(\tfrac55(1,2)=(1,2)\). The distance is \(\|(5,0)-(1,2)\|=\|(4,-2)\|=\sqrt{20}=2\sqrt5\).
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-008 Projecting u=(5,3) onto v=(1,1) and onto 2v=(2,2) gives:
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