4. $$\mathbf { a } = \left( \begin{array} { r } - 3
1
4 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { r } 5
- 2
9 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { r } 8
- 4
3 \end{array} \right)$$ The points \(A , B\) and \(C\) with position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) ，respectively，are 3 vertices of a cube．
（a）Find the volume of the cube． The points \(P , Q\) and \(R\) are vertices of a second cube with \(\overrightarrow { P Q } = \left( \begin{array} { l } 3
4
\alpha \end{array} \right) , \overrightarrow { P R } = \left( \begin{array} { l } 7
1
0 \end{array} \right)\) and \(\alpha\) a positive constant．
（b）Given that angle \(Q P R = 60 ^ { \circ }\) ，find the value of \(\alpha\) ．
（c）Find the length of a diagonal of the second cube．