Given vectors \(\mathbf{v}\) and \(\mathbf{w}\) such that \(\|\mathbf{v}\| = 3,\|\mathbf{w}\| = 7,\) and \(\mathbf{v} \cdot \mathbf{w} = 10,\) then find \(\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\|.\)
+26367 Given vectors \(\mathbf{v}\) and \(\mathbf{w}\) such that \(\|\mathbf{v}\| = 3,\ \|\mathbf{w}\| = 7,\) and \(\mathbf{v} \cdot \mathbf{w} = 10,\) then find \(\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\|.\)
\(\begin{array}{|rcll|} \hline \|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\| &=& \left\| \dfrac{\mathbf{v} \cdot \mathbf{w}}{\left\|\mathbf{w}\right\|^2}\mathbf{w} \right\| \\\\ &=& \left\| \dfrac{10}{7^2}\mathbf{w} \right\| \\\\ &=& \dfrac{10}{7^2} \| \mathbf{w} \| \\\\ &=& \dfrac{10}{7^2} \cdot 7 \\\\ \|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\| &=& \mathbf{\dfrac{10}{7}} \\ \hline \end{array}\)
