9. The line \(L _ { 1 }\) passes through the points \(A ( 1,2 , - 3 )\) and \(B ( - 2,1,0 )\).
- Show that the vector equation of \(L _ { 1 }\) can be written as
$$\mathbf { r } = ( 1 - 3 \lambda ) \mathbf { i } + ( 2 - \lambda ) \mathbf { j } + ( - 3 + 3 \lambda ) \mathbf { k }$$
- Write down the equation of \(L _ { 1 }\) in Cartesian form.
The vector equation of the line \(L _ { 2 }\) is given by \(\mathbf { r } = 2 \mathbf { i } - 4 \mathbf { j } + \mu ( 4 \mathbf { j } + 7 \mathbf { k } )\).
- Show that \(L _ { 1 }\) and \(L _ { 2 }\) do not intersect.
- Find a vector in the direction of the common perpendicular to \(L _ { 1 }\) and \(L _ { 2 }\).