2012 MJC Promo Q6

Relative to the origin \(O\), points \(A\) and \(B\) have position vectors \(4\mathbf{i}+3\mathbf{j}\) and \(5\mathbf{i}+7\mathbf{j}\) respectively. The line \({{l}_{1}}\) passes through point \(B\) and is parallel to the vector \(\mathbf{i}+2\mathbf{j}+\alpha \mathbf{k}\). The line \({{l}_{2}}\) has cartesian equation \(\frac{1-x}{2}=\frac{z-2}{1}\), \(y=4\).

1. Given that \({{l}_{1}}\) and \({{l}_{2}}\) are perpendicular, show that the value of \(\alpha \) is \(2\).[2]
2. Find the length of projection of \(\overrightarrow{BA}\) onto line \({{l}_{1}}\). Hence find the vector \(\overrightarrow{BN}\), where \(N\) is the foot of perpendicular from \(A\) to \({{l}_{1}}\).[4]
3. Point \(A'\) is the reflection of point \(A\) in the line \({{l}_{1}}\) and point \(C\) is such that \(AA'CB\) forms a parallelogram. Find the exact area of the parallelogram \(AA'CB\).[3]