The projection of vector \( \vec{a} \) onto vector \( \vec{b} \) is given by:\[{Proj}_{\vec{b}} (\vec{a}) = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|^2} \vec{b}.\]The projection of vector \( \vec{b} \) onto vector \( \vec{a} \) is given by:\[{Proj}_{\vec{a}} (\vec{b}) = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|^2} \vec{a}.\]These two projections are distinct unless the magnitudes of \( \vec{a} \) and \( \vec{b} \) are equal, i.e., \( \|\vec{a}\| = \|\vec{b}\| \).The angle between \( \vec{a} \) and \( \vec{b} \) is identical to the angle between \( \vec{b} \) and \( \vec{a} \), due to the symmetric nature of the cosine function. Therefore, the reason provided is valid.
Final Answer: \( \boxed{{[(D)] Assertion (A) is false, but Reason (R) is true.}} \)