Exercise · ZW-VECTOR-NORM-010 · ★★★ Advanced
Which statement holds for all vectors \(u,v\) and every real \(c\)?
Choose one option.
Show answer
- A. \(\|u+v\|\le\|u\|+\|v\|\)
Explanation
This is the triangle inequality. Equality needs the vectors to point the same way, and \(\|cv\|=|c|\,\|v\|\) requires the absolute value.
Review the concept: Vector norm
The norm of a vector in \(\mathbb{R}^n\) is a nonnegative number measuring its length: the square root of the vector's dot product with itself.
\[\|x\|=\sqrt{x\cdot x}=\sqrt{x_1^2+\dots+x_n^2}\]
More practice problems: vector norm
- ZW-VECTOR-NORM-001 Compute \|(3,4)\|.
- ZW-VECTOR-NORM-002 Compute \|(1,2,2)\|.
- ZW-VECTOR-NORM-003 Compute \|(2,-3,6)\|.
- ZW-VECTOR-NORM-004 If \|v\|=4, what is \|3v\|?
- ZW-VECTOR-NORM-005 How do \|v\| and \|-v\| compare?
- ZW-VECTOR-NORM-006 Which unit vector points in the direction of (3,-4)?
- ZW-VECTOR-NORM-007 Find k>0 such that \|(k,2k,2k)\|=12.
- ZW-VECTOR-NORM-008 If \|u\|=3 and \|v\|=4, which of these values is impossible for \|u+v\|?