Scalar projection of $$$\left\langle -2, 5, -1\right\rangle$$$ onto $$$\left\langle -10, 7, -1\right\rangle$$$

Calculate the scalar projection of $$$\mathbf{\vec{v}} = \left\langle -2, 5, -1\right\rangle$$$ onto $$$\mathbf{\vec{u}} = \left\langle -10, 7, -1\right\rangle$$$.

The scalar projection is given by $$$\frac{\mathbf{\vec{v}}\cdot \mathbf{\vec{u}}}{\mathbf{\left\lvert\vec{u}\right\rvert}}$$$.

$$$\mathbf{\vec{v}}\cdot \mathbf{\vec{u}} = 56$$$ (for steps, see dot product calculator).

$$$\mathbf{\left\lvert\vec{u}\right\rvert} = 5 \sqrt{6}$$$ (for steps, see vector magnitude calculator).

Thus, the scalar projection is $$$\frac{\mathbf{\vec{v}}\cdot \mathbf{\vec{u}}}{\mathbf{\left\lvert\vec{u}\right\rvert}} = \frac{56}{5 \sqrt{6}} = \frac{28 \sqrt{6}}{15}.$$$

The scalar projection is $$$\frac{28 \sqrt{6}}{15}\approx 4.572380853195266$$$A.