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

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

The vector projection is given by $$$\operatorname{proj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right) = \frac{\mathbf{\vec{v}}\cdot \mathbf{\vec{u}}}{\mathbf{\left\lvert\vec{u}\right\rvert}^{2}} \mathbf{\vec{u}}.$$$

$$$\mathbf{\vec{v}}\cdot \mathbf{\vec{u}} = 4$$$ (for steps, see dot product calculator).

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

Thus, the vector projection is $$$\operatorname{proj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right) = \frac{4}{\left(\sqrt{6}\right)^{2}}\cdot \left\langle 1, 1, 2\right\rangle = \frac{2}{3}\cdot \left\langle 1, 1, 2\right\rangle = \left\langle \frac{2}{3}, \frac{2}{3}, \frac{4}{3}\right\rangle$$$ (for steps, see vector scalar multiplication calculator).

The vector projection is $$$\left\langle \frac{2}{3}, \frac{2}{3}, \frac{4}{3}\right\rangle\approx \left\langle 0.666666666666667, 0.666666666666667, 1.333333333333333\right\rangle.$$$A