Exercise · ZW-ORTHOGONAL-VECTORS-010 · ★★★ Advanced
Let \(u\neq0\) in \(\mathbb{R}^3\). The set of vectors orthogonal to \(u\) is:
Choose one option.
Show answer
- B. a plane through the origin
Explanation
The condition \(u\cdot x=0\) is one homogeneous linear equation in three unknowns: it defines a plane through the origin (the origin satisfies it).
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-007 For any (a,b)≠(0,0), a vector orthogonal to (a,b) in ℝ^2 is:
- ZW-ORTHOGONAL-VECTORS-008 Let u,v be nonzero with u· v=0. Then: