Vector projection of $$$\left\langle -4, 1, 3\right\rangle$$$ onto $$$\left\langle 5, 4, 4\right\rangle$$$

Calculate the vector projection of $$$\mathbf{\vec{v}} = \left\langle -4, 1, 3\right\rangle$$$ onto $$$\mathbf{\vec{u}} = \left\langle 5, 4, 4\right\rangle$$$.

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).

$$$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{57}$$$ (for steps, see vector magnitude calculator).

Thus, the vector projection is $$$\operatorname{proj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right) = \frac{-4}{\left(\sqrt{57}\right)^{2}}\cdot \left\langle 5, 4, 4\right\rangle = \left(- \frac{4}{57}\right)\cdot \left\langle 5, 4, 4\right\rangle = \left\langle - \frac{20}{57}, - \frac{16}{57}, - \frac{16}{57}\right\rangle$$$ (for steps, see vector scalar multiplication calculator).

The vector projection is $$$\left\langle - \frac{20}{57}, - \frac{16}{57}, - \frac{16}{57}\right\rangle\approx \left\langle -0.350877192982456, -0.280701754385965, -0.280701754385965\right\rangle.$$$A