Scalar projection of $$$\left\langle 5, -1, 2\right\rangle$$$ onto $$$\left\langle 4, 4, -3\right\rangle$$$

Calculate the scalar projection of $$$\mathbf{\vec{v}} = \left\langle 5, -1, 2\right\rangle$$$ onto $$$\mathbf{\vec{u}} = \left\langle 4, 4, -3\right\rangle$$$.

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

$$$\mathbf{\vec{v}}\cdot \mathbf{\vec{u}} = 10$$$ (for steps, see dot product calculator).

$$$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{41}$$$ (for steps, see vector magnitude calculator).

Thus, the scalar projection is $$$\frac{\mathbf{\vec{v}}\cdot \mathbf{\vec{u}}}{\mathbf{\left\lvert\vec{u}\right\rvert}} = \frac{10}{\sqrt{41}} = \frac{10 \sqrt{41}}{41}.$$$

The scalar projection is $$$\frac{10 \sqrt{41}}{41}\approx 1.561737618886061$$$A.