2011 MI C2 MYE P2 Q4 [Modified]

The line \({{l}_{1}}\) passes through the points \(A\) and \(B\) which have position vectors \(2\mathbf{i}-3\mathbf{j}+\mathbf{k}\) and \(3\mathbf{i}-\mathbf{j}\) respectively. The line \({{l}_{2}}\) has Cartesian equation \(2x=y+7=\frac{z-3}{2}\). Find

1. the vector equations of \({{l}_{1}}\) and \({{l}_{2}}\).
2. the acute angle between the two lines,
3. the position vector of the point of intersection between \({{l}_{1}}\) and \({{l}_{2}}\).

The line \({{l}_{3}}\) contains the point \(B\). It also is perpendicular to and intersects \({{l}_{2}}\).

4. Find the coordinates of the point of intersection between \({{l}_{2}}\) and \({{l}_{3}}\) and the Cartesian equation of \({{l}_{3}}\).
5. Hence, find the area of the triangle bounded by the three lines.