4 The plane $\Pi$ contains the lines $\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )$ and $\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )$.\\
(a) Find a Cartesian equation of $\Pi$, giving your answer in the form $a x + b y + c z = d$.\\

The line $l$ passes through the point $P$ with position vector $2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }$ and is parallel to the vector $\mathbf { j } + \mathbf { k }$.\\
(b) Find the acute angle between $I$ and $\Pi$.\\

(c) Find the position vector of the foot of the perpendicular from $P$ to $\Pi$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2022 Q4}}