5 The lines $l _ { 1 }$ and $l _ { 2 }$ have equations $\mathbf { r } = 3 \mathbf { i } + 3 \mathbf { k } + \lambda ( \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k } )$ and $\mathbf { r } = 3 \mathbf { i } - 5 \mathbf { j } - 6 \mathbf { k } + \mu ( 5 \mathbf { j } + 6 \mathbf { k } )$ respectively.\\
(a) Find the shortest distance between $l _ { 1 }$ and $l _ { 2 }$.\\

The plane $\Pi$ contains $l _ { 1 }$ and is parallel to the vector $\mathbf { i } + \mathbf { k }$.\\
(b) Find the equation of $\Pi$, giving your answer in the form $a x + b y + c z = d$.\\

(c) Find the acute angle between $l _ { 2 }$ and $\Pi$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2020 Q5}}