2025 TMJC P2 Q5

Referred to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively where \(\mathbf{a}\) is a non-zero vector and \(\mathbf{b} \neq \mathbf{c}\).

1. Given that \(\mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c}\), show that \(\mathbf{b} - \mathbf{c} = \lambda \mathbf{a}\), where \(\lambda\) is a non-zero constant.[2]

It is given that the area of the trapezium \(OABC\) is \(10\) units\({}^2\), \(|\mathbf{a}| = 3\), \(|\mathbf{c}| = 5\) and the angle between \(\mathbf{a}\) and \(\mathbf{c}\) is \(30{}^\circ \).

2. Show that \(BC = 5\) units.[3]
3. State the value of \(k\) such that \(\overrightarrow{OA} = k\overrightarrow{CB}\), where \(k\) is a positive constant.[1]
4. The line \(OC\) meets the line \(AB\) at point \(D\). Find \(\overrightarrow{OD}\) in terms of \(\mathbf{c}\).[5]