Exercise · ZW-ORTHOGONAL-PROJECTION-010 · ★★★ Advanced
If \(\operatorname{proj}_v u=(1,2)\) and \(\operatorname{proj}_v w=(3,-1)\), then \(\operatorname{proj}_v(u+w)\) is:
Choose one option.
Show answer
- B. \((4,1)\)
Explanation
Projection is a linear operation: \(\operatorname{proj}_v(u+w)=\operatorname{proj}_v u+\operatorname{proj}_v w=(4,1)\).
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: