JPJC Vectors Tutorial Q2

The points \(A\), \(B\) and \(C\) have position vectors \(\alpha \mathbf{i}-\mathbf{j}+2\mathbf{k}\), \(2\mathbf{i}-4\mathbf{j}\) and \(-\mathbf{i}-7\mathbf{j}+\beta \mathbf{k}\) respectively where \(\alpha\) and \(\beta\) are constants.

1. Given that \(A\), \(B\) and \(C\) are collinear, find the values of \(\alpha\) and \(\beta\) are constants.[4]
2. The point \(P\) lies on \(OB\) such that \(\overrightarrow{OP}=k\overrightarrow{OB}\) , where \(0 < k < 1\). \(Q\) is the midpoint of \(BC\) and the length of projection of \(\overrightarrow{PQ}\) onto the line \(OB\) is \(\sqrt{5}\) units. Find the value of \(k\) .[5]