2019 TJC Promo Q2

Interpret geometrically the vector equation \(\mathbf{r}=\mathbf{a}+\mu \mathbf{m}\) where \(\mathbf{a}\) and \(\mathbf{m}\) are constant vectors and \(\mu \) is a parameter.[2]

Referred to the origin \(O\), the points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively, such that \(\mathbf{a}\) and \(\mathbf{b}\) are non-parallel vectors. The point \(C\) lies on the line \(AB\) such that the area of the triangle \(OBC\) is \(6\) units\({^2}\). Given that \(\mathbf{a}\) is a unit vector, \(\left| \mathbf{b} \right|=4\) and the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(30{}^\circ \), find the possible vectors of \(C\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).[6]