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# help

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Given vectors $$\mathbf{v}$$ and $$\mathbf{w}$$ such that $$\|\mathbf{v}\| = 3,\|\mathbf{w}\| = 7,$$ and $$\mathbf{v} \cdot \mathbf{w} = 10,$$ then find $$\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\|.$$

Sep 3, 2019

#1
+23357
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Given vectors $$\mathbf{v}$$ and $$\mathbf{w}$$ such that $$\|\mathbf{v}\| = 3,\ \|\mathbf{w}\| = 7,$$ and $$\mathbf{v} \cdot \mathbf{w} = 10,$$ then find $$\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\|.$$

$$\begin{array}{|rcll|} \hline \|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\| &=& \left\| \dfrac{\mathbf{v} \cdot \mathbf{w}}{\left\|\mathbf{w}\right\|^2}\mathbf{w} \right\| \\\\ &=& \left\| \dfrac{10}{7^2}\mathbf{w} \right\| \\\\ &=& \dfrac{10}{7^2} \| \mathbf{w} \| \\\\ &=& \dfrac{10}{7^2} \cdot 7 \\\\ \|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\| &=& \mathbf{\dfrac{10}{7}} \\ \hline \end{array}$$

Sep 4, 2019