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

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

respectively.\\
(i) Show that $l _ { 1 }$ and $l _ { 2 }$ intersect.\\
(ii) Find the perpendicular distance from the point $P$ whose position vector is $3 \mathbf { i } - 5 \mathbf { j } + 6 \mathbf { k }$ to the plane containing $l _ { 1 }$ and $l _ { 2 }$.\\
(iii) Find the perpendicular distance from $P$ to $l _ { 1 }$.

\hfill \mbox{\textit{CAIE FP1 2010 Q7}}