Exercise · ZW-ORTHOGONAL-VECTORS-007 · ★★☆ Intermediate
For any \((a,b)\neq(0,0)\), a vector orthogonal to \((a,b)\) in \(\mathbb{R}^2\) is:
Choose one option.
Show answer
- D. \((-b,a)\)
Explanation
\((a,b)\cdot(-b,a)=-ab+ab=0\). The others give \(2ab\), \(a^2-b^2\) and \(-(a^2+b^2)\), which do not vanish in general.
Review the concept: Orthogonal vectors
Two vectors of \(\mathbb{R}^n\) are orthogonal when their dot product is zero; geometrically, they are perpendicular.
\[u\perp v\iff u\cdot v=0\]
More practice problems: orthogonal vectors
- ZW-ORTHOGONAL-VECTORS-001 Which of the following pairs of vectors is orthogonal?
- ZW-ORTHOGONAL-VECTORS-002 For which value of k are (k,3) and (2,-4) orthogonal?
- ZW-ORTHOGONAL-VECTORS-003 Find k so that (1,k,2) and (3,-1,-1) are orthogonal.
- ZW-ORTHOGONAL-VECTORS-004 The zero vector of ℝ^n is orthogonal to:
- ZW-ORTHOGONAL-VECTORS-005 Which of these vectors is orthogonal to (1,-2,2)?
- ZW-ORTHOGONAL-VECTORS-006 If u and v are orthogonal, \|u\|=3 and \|v\|=4, what is \|u+v\|?
- ZW-ORTHOGONAL-VECTORS-008 Let u,v be nonzero with u· v=0. Then:
- ZW-ORTHOGONAL-VECTORS-009 Which vector is orthogonal to both (1,0,1) and (0,1,1)?