2022 NJC P1 Q3

Relative to the origin \(O\), points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(3\mathbf{a}-4\mathbf{b}\) respectively, where \(\mathbf{a}\) and \(\mathbf{b}\) are non-zero and non-parallel.

1. Given that \(OAB\) is an equilateral triangle, show that \(\mathbf{a}\cdot \mathbf{b}=k{{\left| \mathbf{a} \right|}^{2}}\), where \(k\) is a constant to be determined.[2]
2. Deduce whether \(OA\) and \(AC\) are perpendicular.[2]
3. Interpret geometrically the vector equation \(\mathbf{r}\times \left( \mathbf{b}-\mathbf{a} \right)=\mathbf{a}\times \left( \mathbf{b}-\mathbf{a} \right)\).[2]