10 The line $l _ { 1 }$ is parallel to the vector $a \mathbf { i } - \mathbf { j } + \mathbf { k }$, where $a$ is a constant, and passes through the point whose position vector is $9 \mathbf { j } + 2 \mathbf { k }$. The line $l _ { 2 }$ is parallel to the vector $- a \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }$ and passes through the point whose position vector is $- 6 \mathbf { i } - 5 \mathbf { j } + 10 \mathbf { k }$.\\
(i) It is given that $l _ { 1 }$ and $l _ { 2 }$ intersect.\\
(a) Show that $a = - \frac { 6 } { 13 }$.\\

(b) Find a cartesian equation of the plane containing $l _ { 1 }$ and $l _ { 2 }$.\\

(ii) Given instead that the perpendicular distance between $l _ { 1 }$ and $l _ { 2 }$ is $3 \sqrt { } ( 30 )$, find the value of $a$.\\

\hfill \mbox{\textit{CAIE FP1 2018 Q10}}