Exercise · ZW-VECTOR-IN-RN-010 · ★★★ Advanced
For \(u=(1,2)\) and \(v=(3,-1)\), compute \(\|u+v\|^2\).
Choose one option.
Show answer
- C. \(17\)
Explanation
\(u+v=(4,1)\), so \(\|u+v\|^2=16+1=17\). The value 15 is \(\|u\|^2+\|v\|^2\): it forgets the cross term \(2u\cdot v=2\).
Review the concept: Vector in ℝⁿ
A vector in \(\mathbb{R}^n\) is an ordered list of \(n\) real numbers, which geometrically looks like an arrow from the origin.
\[x=(x_1,\dots,x_n),\quad cx=(cx_1,\dots,cx_n)\]
More practice problems: vector in ℝⁿ
- ZW-VECTOR-IN-RN-001 Compute u+v for u=(1,2,3) and v=(4,-1,0).
- ZW-VECTOR-IN-RN-002 Compute 3·(2,-1,4).
- ZW-VECTOR-IN-RN-003 If u=(1,0,-2) and v=(3,1,1), compute 2u-v.
- ZW-VECTOR-IN-RN-004 Compute the norm \|x\| of x=(3,4,12).
- ZW-VECTOR-IN-RN-005 Compute the dot product (1,2,3)·(4,-5,6).
- ZW-VECTOR-IN-RN-006 Which space ℝ^n does (1,0,-2,5) belong to?
- ZW-VECTOR-IN-RN-007 If (x,2,1)+(3,y,-1)=(5,0,0), what is the pair (x,y)?
- ZW-VECTOR-IN-RN-008 Which scalars (c_1,c_2) satisfy c_1(1,2)+c_2(3,-1)=(5,3)?