Vector projection of $$$\left\langle 4, -2, -4\right\rangle$$$ onto $$$\left\langle 1, 2, -2\right\rangle$$$

Calculate the vector projection of $$$\mathbf{\vec{v}} = \left\langle 4, -2, -4\right\rangle$$$ onto $$$\mathbf{\vec{u}} = \left\langle 1, 2, -2\right\rangle$$$.

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

$$$\mathbf{\vec{v}}\cdot \mathbf{\vec{u}} = 8$$$ (for steps, see dot product calculator).

$$$\mathbf{\left\lvert\vec{u}\right\rvert} = 3$$$ (for steps, see vector magnitude calculator).

Thus, the vector projection is $$$\operatorname{proj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right) = \frac{8}{3^{2}}\cdot \left\langle 1, 2, -2\right\rangle = \frac{8}{9}\cdot \left\langle 1, 2, -2\right\rangle = \left\langle \frac{8}{9}, \frac{16}{9}, - \frac{16}{9}\right\rangle$$$ (for steps, see vector scalar multiplication calculator).

The vector projection is $$$\left\langle \frac{8}{9}, \frac{16}{9}, - \frac{16}{9}\right\rangle\approx \left\langle 0.888888888888889, 1.777777777777778, -1.777777777777778\right\rangle.$$$A