Scalar projection of $$$\left\langle 5, -3, -1\right\rangle$$$ onto $$$\left\langle -1, 2, 5\right\rangle$$$

The calculator will find the scalar projection of the vector $$$\left\langle 5, -3, -1\right\rangle$$$ onto the vector $$$\left\langle -1, 2, 5\right\rangle$$$, with steps shown.

Related calculator: Vector Projection Calculator

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Calculate the scalar projection of $$$\mathbf{\vec{v}} = \left\langle 5, -3, -1\right\rangle$$$ onto $$$\mathbf{\vec{u}} = \left\langle -1, 2, 5\right\rangle$$$.


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 $$$\frac{\mathbf{\vec{v}}\cdot \mathbf{\vec{u}}}{\mathbf{\left\lvert\vec{u}\right\rvert}} = \frac{-16}{\sqrt{30}} = - \frac{8 \sqrt{30}}{15}.$$$


The scalar projection is $$$- \frac{8 \sqrt{30}}{15}\approx -2.921186973360886$$$A.