6 Let $t$ be a positive constant.\\
The line $l _ { 1 }$ passes through the point with position vector $t \mathbf { i } + \mathbf { j }$ and is parallel to the vector $- 2 \mathbf { i } - \mathbf { j }$. The line $l _ { 2 }$ passes through the point with position vector $\mathbf { j } + t \mathbf { k }$ and is parallel to the vector $- 2 \mathbf { j } + \mathbf { k }$.

It is given that the shortest distance between the lines $l _ { 1 }$ and $l _ { 2 }$ is $\sqrt { \mathbf { 2 1 } }$.\\
(a) Find the value of $t$.\\

The plane $\Pi _ { 1 }$ contains $l _ { 1 }$ and is parallel to $l _ { 2 }$.\\
(b) Write down an equation of $\Pi _ { 1 }$, giving your answer in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }$.\\

The plane $\Pi _ { 2 }$ has Cartesian equation $5 x - 6 y + 7 z = 0$.\\
(c) Find the acute angle between $l _ { 2 }$ and $\Pi _ { 2 }$.\\

(d) Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2021 Q6}}