Find

- the equation (in any form) of the line \(AB\),
- the shortest distance between \(AB\) and \(CD\),
- the equation (in any form) of the plane \(ABC\),
- the angle between the planes \(ABC\) and \(ABD\).

The position vectors of four points \(A\), \(B\), \(C\), \(D\) relative to an origin \(O\) are given below. The vectors \(\mathbf{i}\), \(\mathbf{j}\), \(\mathbf{k}\) are mutually perpendicular unit vectors.
\[\begin{align*}
A \colon\;& 2\mathbf{i} + 3\mathbf{j} + \mathbf{k}, \\
B \colon\;& \phantom{2}\mathbf{i} + \phantom{3}\mathbf{j} - \mathbf{k}, \\
C \colon\;& \phantom{2}\mathbf{i} \phantom{{}+{}3\mathbf{i}} + \mathbf{k}, \\
D \colon\;& \phantom{2\mathbf{i}{}+{}} 3\mathbf{j}.\phantom{{}+{}\mathbf{k}}
\end{align*}\]

Find

- the equation (in any form) of the line \(AB\),
- the shortest distance between \(AB\) and \(CD\),
- the equation (in any form) of the plane \(ABC\),
- the angle between the planes \(ABC\) and \(ABD\).