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

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

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

$$$\mathbf{\vec{v}}\cdot \mathbf{\vec{u}} = 2$$$ (for steps, see dot product calculator).

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

Thus, the scalar projection is $$$\frac{\mathbf{\vec{v}}\cdot \mathbf{\vec{u}}}{\mathbf{\left\lvert\vec{u}\right\rvert}} = \frac{2}{\sqrt{6}} = \frac{\sqrt{6}}{3}.$$$

The scalar projection is $$$\frac{\sqrt{6}}{3}\approx 0.816496580927726$$$A.