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

 Feb 23, 2020
 #1
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Projection is linear, so the answer is 2*3*(4,-7) = (24,-42).

 Feb 24, 2020
 #2
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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
\(\operatorname{proj}_{-2 {\mathbf{w}}} (3 {\mathbf{v}})\).

 

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

 

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

 

laugh

 Feb 24, 2020

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