5 The points $A , B , C$ have position vectors

$$2 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } , \quad 2 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } , \quad - 3 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }$$

respectively, relative to the origin $O$.\\
(a) Find the equation of the plane $A B C$, giving your answer in the form $a x + b y + c z = d$.\\

The point $D$ has position vector $2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }$.\\
(b) Find the perpendicular distance from $D$ to the plane $A B C$.\\

(c) Find the shortest distance between the lines $A B$ and $C D$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2024 Q5}}