Let $$\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$$ and $$\vec{b} = \hat{i} + 3\hat{j} + 5\hat{k}$$ be two vectors. Then which one of the following statements is TRUE?

Given $$\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$$ and $$\vec{b} = \hat{i} + 3\hat{j} + 5\hat{k}$$. First, compute their dot product: $$\vec{a} \cdot \vec{b} = (5)(1) + (-1)(3) + (-3)(5) = 5 - 3 - 15 = -13$$. Next, the magnitude of $$\vec{b}$$ is $$|\vec{b}| = \sqrt{1^2 + 3^2 + 5^2} = \sqrt{1 + 9 + 25} = \sqrt{35}$$. The scalar projection of $$\vec{a}$$ on $$\vec{b}$$ is then $$\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} = \frac{-13}{\sqrt{35}}$$, which is negative, so the projection vector points in the direction opposite to $$\vec{b}$$.

Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{-13}{\sqrt{35}}$$ and the direction of the projection vector is opposite to the direction of $$\vec{b}$$.