$$|\vec{v}| = \sqrt{3^2 + 0^2 + 4^2} = \sqrt{25} = 5$$

$$\hat{v} = \frac{1}{|\vec{v}|} \vec{v} = \frac{1}{5}(3, 0, 4) = \left(\frac{3}{5}, 0, \frac{4}{5}\right)$$

$$\vec{u} \cdot \hat{v} = (2, -1, 3) \cdot \left(\frac{3}{5}, 0, \frac{4}{5}\right)$$ $$= 2\left(\frac{3}{5}\right) + (-1)(0) + 3\left(\frac{4}{5}\right)$$ $$= \frac{6}{5} + 0 + \frac{12}{5} = \frac{18}{5}$$

Vector projection of \(\vec{u}\) along \(\hat{v}\) is:

$$(\vec{u} \cdot \hat{v})\hat{v} = \frac{18}{5}\left(\frac{3}{5}, 0, \frac{4}{5}\right)$$ $$= \left(\frac{18}{5} \cdot \frac{3}{5}, \frac{18}{5} \cdot 0, \frac{18}{5} \cdot \frac{4}{5}\right)$$ $$= \left(\frac{54}{25}, 0, \frac{72}{25}\right)$$