Scalar projection of $$$\left\langle 5, -3, -1\right\rangle$$$ onto $$$\left\langle -1, 2, 5\right\rangle$$$
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Your Input
Calculate the scalar projection of $$$\mathbf{\vec{v}} = \left\langle 5, -3, -1\right\rangle$$$ onto $$$\mathbf{\vec{u}} = \left\langle -1, 2, 5\right\rangle$$$.
Solution
The scalar projection is given by $$$\frac{\mathbf{\vec{v}}\cdot \mathbf{\vec{u}}}{\mathbf{\left\lvert\vec{u}\right\rvert}}$$$.
$$$\mathbf{\vec{v}}\cdot \mathbf{\vec{u}} = -16$$$ (for steps, see dot product calculator).
$$$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{30}$$$ (for steps, see vector magnitude calculator).
Thus, the scalar projection is $$$\frac{\mathbf{\vec{v}}\cdot \mathbf{\vec{u}}}{\mathbf{\left\lvert\vec{u}\right\rvert}} = \frac{-16}{\sqrt{30}} = - \frac{8 \sqrt{30}}{15}.$$$
Answer
The scalar projection is $$$- \frac{8 \sqrt{30}}{15}\approx -2.921186973360886$$$A.