2024 ACJC P2 Q4

The origin \(O\) and the points \(A\), \(B\) and \(C\) lie in the same plane, where \(\overrightarrow{OA} = \mathbf{a}\), \(\overrightarrow{OB} = \mathbf{b}\) and \(\overrightarrow{OC} = \mathbf{c}\). It is given that \(\angle AOC = \angle BOC = \theta\), where \(\theta\) is an acute angle.

1. Show that \(\mathbf{c} \cdot \mathbf{\hat{a}} = \mathbf{c} \cdot \mathbf{\hat{b}}\) where \(\mathbf{\hat{a}}\) and \(\mathbf{\hat{b}}\) are unit vectors in the directions of vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively.[1]
2. If \(\mathbf{c}\) can be written as \(m\mathbf{\hat{a}} + n\mathbf{\hat{b}}\), where \(m\) and \(n\) are constants, use the result from (a) to show that \(m = n\).[3]
3. Write down the equation of the line passing through the points \(A\) and \(B\).[1]
4. Given that \(|\mathbf{a}| = 3\), \(|\mathbf{b}| = 2\) and \(m = n\), show that the position vector of the point of intersection of the line passing through \(A\) and \(B\) and the line passing through \(O\) and \(C\) is \(t\left(\mathbf{\hat{a}} + \mathbf{\hat{b}}\right)\), where \(t\) is a constant to be determined.[3]