Find the vector projection of $\mathbf{b}=(4,-5,2)$ onto $\mathbf{a}=(3,1,0)$.

Step-by-step solution

1. Step 1: Use the projection formula: ${\mathrm{proj}}_{\mathbf{a}}\mathbf{b}=\frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{a}{|}^2}\mathbf{a}$.
2. Step 2: Compute the dot product: $\mathbf{a}\cdot \mathbf{b}=3\cdot 4+1\cdot (-5)+0\cdot 2=12-5=7$.
3. Step 3: Compute $|\mathbf{a}{|}^2=3^2+1^2+0^2=9+1=10$.
4. Step 4: Calculate the scalar factor: $\frac{7}{10}$.
5. Step 5: Multiply this factor by $\mathbf{a}$: $\frac{7}{10}(3,1,0)=(\frac{21}{10},\frac{7}{10},0)$.
6. Step 6: Conclude the projection is $(\frac{21}{10},\frac{7}{10},0)$.

Concept tested: Vector projection

Published 18 September 2026 · How this solution was prepared and checked