Using vector methods, prove that the angle subtended by a diameter of a circle at any point on the circumference is a right angle.

Given vectors $\vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$: (a) Find the dot product $\vec{a} \cdot \vec{b}$ and the angle between them. (b) Find the projection of $\vec{a}$ on $\vec{b}$. (c) Find the cross product $\vec{a} \times \vec{b}$ and the unit vector perpendicular to both $\vec{a}$ and $\vec{b}$.