Relative to an origin $O$, the point $A$ has position vector $4i + 7j - pk$ and the point $B$ has position vector $8i - j - pk$, where $p$ is a constant.
a) Find $\overrightarrow {OA} \,.\,\,\overrightarrow {OB} $.
b) Hence show that there are no real values of $p$ for which $OA$ and $OB$ are perpendicular to each other.
c) Find the values of $p$ for which angle $AOB = {60^ \circ }$
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