ASRJC Vectors Assignment Q1

The position vectors of points \(A\), \(B\) and \(C\) with respect to the origin \(O\) are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively.

1. It is given that \(\overrightarrow{AC}=k\mathbf{a}-3\mathbf{b}\). State the value of \(k\) such that \(A\), \(B\) and \(C\) are collinear.
2. Using the value of \(k\) found in (i) and given that \(\left| \mathbf{a} \right|=2\), \(\mathbf{a}\cdot \mathbf{b}=3\), find the length of projection of \(\overrightarrow{OC}\) on \(\overrightarrow{OA}\).