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?

$$\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$$ and $$\vec{b} = \hat{i} + 3\hat{j} + 5\hat{k}$$. Determine which statement about the projection of $$\vec{a}$$ on $$\vec{b}$$ is true.

The dot product $$\vec{a} \cdot \vec{b}$$ is calculated as $$(5)(1) + (-1)(3) + (-3)(5) = 5 - 3 - 15 = -13$$. 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 $$\text{proj} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} = \frac{-13}{\sqrt{35}}$$.

The projection vector is given by $$\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\,\vec{b} = \frac{-13}{35}\vec{b}$$, and since the scalar multiplier is negative ($$-13/35 < 0$$), the projection vector points in the direction opposite to $$\vec{b}$$.

Option A states that the projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{-13}{\sqrt{35}}$$ and that the projection vector points opposite to $$\vec{b}$$, which matches our calculation exactly.

Answer: Option A