4 The lines $l _ { 1 }$ and $l _ { 2 }$ have equations

$$\mathbf { r } = - 2 \mathbf { i } - 3 \mathbf { j } - 5 \mathbf { k } + \lambda ( - 4 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )$$

respectively.\\
(a) Find the shortest distance between $l _ { 1 }$ and $l _ { 2 }$.\\

The plane $\Pi$ contains $l _ { 1 }$ and the point with position vector $- \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }$.\\
(b) Find an equation of $\Pi$, giving your answer in the form $a x + b y + c z = d$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2023 Q4}}