arrow_back Let $\mathbf{u}=(2,-1,3)$ and $\mathbf{a}=(4,-1,2)$. Find the vector component of $\mathbf{u}$ along a and the vector component of $\mathbf{u}$ orthogonal to a.

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Let $\mathbf{u}=(2,-1,3)$ and $\mathbf{a}=(4,-1,2)$. Find the vector component of $\mathbf{u}$ along a and the vector component of $\mathbf{u}$ orthogonal to a.

\begin{aligned} &\mathbf{u} \cdot \mathbf{a}=(2)(4)+(-1)(-1)+(3)(2)=15 \\ &\|\mathbf{a}\|^{2}=4^{2}+(-1)^{2}+2^{2}=21 \end{aligned} Thus the vector component of $\mathbf{u}$ along $\mathbf{a}$ is $$\operatorname{proj}_{\mathbf{a}} \mathbf{u}=\frac{\mathbf{u} \cdot \mathbf{a}}{\|\mathbf{a}\|^{2}} \mathbf{a}=\frac{15}{21}(4,-1,2)=\left(\frac{20}{7},-\frac{5}{7}, \frac{10}{7}\right)$$ and the vector component of $\mathbf{u}$ orthogonal to $\mathbf{a}$ is $$\mathbf{u}-\operatorname{proj}_{\mathbf{a}} \mathbf{u}=(2,-1,3)-\left(\frac{20}{7},-\frac{5}{7}, \frac{10}{7}\right)=\left(-\frac{6}{7},-\frac{2}{7}, \frac{11}{7}\right)$$
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