Find the vector projection of $\mathbf{b}$ onto $\mathbf{a}$ where $\mathbf{a}=2\mathbf{i}+3\mathbf{j}-\mathbf{k}$ and $\mathbf{b}=-\mathbf{i}+4\mathbf{j}+2\mathbf{k}$.

Step-by-step solution

1. Use $displaystyleoperatorname{proj}_{\mathbf{a}}\mathbf{b}=\frac{\mathbf{a}cdotmathbfb}{\mathbf{a}cdotmathbfa};\mathbf{a}$.
2. Compute $\mathbf{a}cdotmathbfb=8$ (same as in previous question).
3. Compute $\mathbf{a}cdotmathbfa=2^2+3^2+(-1{)}^2=14$.
4. Find scalar coefficient: $\frac{8}{14}=\frac{4}{7}$.
5. Multiply each component of $\mathbf{a}$ by $\frac{4}{7}$: $\frac{4}{7}(2\mathbf{i}+3\mathbf{j}-\mathbf{k})=\frac{8}{7}\mathbf{i}+\frac{12}{7}\mathbf{j}-\frac{4}{7}\mathbf{k}$.
6. Thus the projection is $\frac{8}{7}\mathbf{i}+\frac{12}{7}\mathbf{j}-\frac{4}{7}\mathbf{k}$.

Concept tested: Vector projection

Published 31 August 2026 · Last updated 16 September 2026 · How this solution was prepared and checked

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