Scalar projection of $$$\left\langle 3, -2, -5\right\rangle$$$ onto $$$\left\langle -1, -5, 0\right\rangle$$$

Calculate the scalar projection of $$$\mathbf{\vec{v}} = \left\langle 3, -2, -5\right\rangle$$$ onto $$$\mathbf{\vec{u}} = \left\langle -1, -5, 0\right\rangle$$$.

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

$$$\mathbf{\vec{v}}\cdot \mathbf{\vec{u}} = 7$$$ (for steps, see dot product calculator).

$$$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{26}$$$ (for steps, see vector magnitude calculator).

Thus, the scalar projection is $$$\frac{\mathbf{\vec{v}}\cdot \mathbf{\vec{u}}}{\mathbf{\left\lvert\vec{u}\right\rvert}} = \frac{7}{\sqrt{26}} = \frac{7 \sqrt{26}}{26}.$$$

The scalar projection is $$$\frac{7 \sqrt{26}}{26}\approx 1.372812945967288$$$A.