Exercise · ZW-VECTOR-NORM-009 · ★★★ Advanced
If \(\|u\|=2\), \(\|v\|=3\) and \(u\cdot v=-1\), what is \(\|u+v\|\)?
Choose one option.
Show answer
- C. \(\sqrt{11}\)
Explanation
\(\|u+v\|^2=\|u\|^2+2\,u\cdot v+\|v\|^2=4-2+9=11\). Adding the norms (5) ignores the cross term.
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\|?