Determine the vector projection of $\mathbf{b}=(4,6,-2)$ onto $\mathbf{a}=(3,-1,2)$.

Step-by-step solution

1. Compute $\mathbf{a}\cdot \mathbf{b}=3\cdot 4+(-1)\cdot 6+2\cdot (-2)=12-6-4=2$.
2. Compute $\Vert \mathbf{a}{\Vert }^2=3^2+(-1{)}^2+2^2=9+1+4=14$.
3. Find the scalar factor: $\frac{\mathbf{a}\cdot \mathbf{b}}{\Vert \mathbf{a}{\Vert }^2}=\frac{2}{14}=\frac{1}{7}$.
4. Multiply this scalar by $\mathbf{a}$ to get the projection: $\frac{1}{7}(3,-1,2)=(3/7,-1/7,2/7)$.
5. Thus the projection vector is $\left(\frac{3}{7},-\frac{1}{7},\frac{2}{7}\right)$.

Concept tested: Vector projection

Published 12 September 2026 · How this solution was prepared and checked