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Let $\mathbf{v}$ and $\mathbf{w}$ be vectors such that \[\operatorname{proj}_{\bold{w}}( \bold{v} )= \begin{pmatrix} 4 \\ -7 \end{pmatrix}.\

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Let v and w​ be vectors such that $$\operatorname{proj}_{{w}}( {v} )= \begin{pmatrix} 4 \\ -7 \end{pmatrix}$$
Find $$\operatorname{proj}_{-2 {w}} (3 {v})$$.

Feb 8, 2019

1+0 Answers

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Let

$$\mathbf{v} \text{ and } \mathbf{w} \text{ be vectors such that } \operatorname{proj}_{\mathbf{w}}( \mathbf{v} )= \begin{pmatrix} 4 \\ -7 \end{pmatrix}.$$

Find

$$\operatorname{proj}_{-2 {\mathbf{w}}} (3 {\mathbf{v}}).$$

$$\begin{array}{|rcll|} \hline \vec{p} &=& \operatorname{proj}_{\mathbf{w}}( \mathbf{v} ) \\\\ &=& \dbinom{4}{-7} \\\\ &=& \dfrac{\vec{w}\vec{v}}{w^2}\vec{w} \\ \hline \vec{P} &=& \operatorname{proj}_{\mathbf{-2w}}( \mathbf{3v} ) \\\\ &=& \dfrac{(-2\vec{w})(3\vec{v})}{|-2w|^2}(-2\vec{w}) \\\\ &=& 6\dfrac{\vec{w}\vec{v}}{2w^2}\vec{w} \\\\ &=& 3\dfrac{\vec{w}\vec{v}}{w^2}\vec{w} \\\\ &=& 3\vec{p} \\\\ &=& 3\dbinom{4}{-7} \\\\ &=& \dbinom{12}{-21} \\ \hline \end{array}$$

$$\operatorname{proj}_{-2 {\mathbf{w}}} (3 {\mathbf{v}}) = \dbinom{12}{-21}.$$

Feb 8, 2019