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 \(ABD\).
Combining Vectors
Question
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.
\[\begin{align*}
A \colon\;& 2\mathbf{i} + 3\mathbf{j} + \mathbf{k}, \\
B \colon\;& \phantom{2}\mathbf{i} + \phantom{3}\mathbf{j} - \mathbf{k}, \\
C \colon\;& \phantom{2}\mathbf{i} \phantom{{}+{}3\mathbf{i}} + \mathbf{k}, \\
D \colon\;& \phantom{2\mathbf{i}{}+{}} 3\mathbf{j}.\phantom{{}+{}\mathbf{k}}
\end{align*}\]
