6. Two submarines are travelling in straight lines through the ocean. Relative to a fixed origin, the vector equations of the two lines, \(l _ { 1 }\) and \(l _ { 2 }\), along which they travel are
$$\begin{aligned}
\mathbf { r } & = 3 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } )
\text { and } \mathbf { r } & = 9 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } + \mu ( 4 \mathbf { i } + \mathbf { j } - \mathbf { k } ) ,
\end{aligned}$$
where \(\lambda\) and \(\mu\) are scalars.
- Show that the submarines are moving in perpendicular directions.
- Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\), find the position vector of \(A\).
The point \(b\) has position vector \(10 \mathbf { j } - 11 \mathbf { k }\).
- Show that only one of the submarines passes through the point \(B\).
- Given that 1 unit on each coordinate axis represents 100 m , find, in km , the distance \(A B\).