2025 CJC Promo Q6

The diagram shows a unit circle with centre \(O\) and diameter \(AB\). The point \(C\) lies on the circumference of the circle. The position vectors of \(A\), \(B\) and \(C\) are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively.

1. By considering an appropriate scalar product, show that \(AC\) and \(BC\) are perpendicular.[4]
2. Give a geometrical interpretation of \(|\mathbf{a}\cdot(\mathbf{c}-\mathbf{a})|\).[1]

The point \(D\) is such that \(ABCD\) is a parallelogram. The point \(M\) is the mid-point of \(AD\) and the point \(R\) lies on \(OA\) produced such that \(OA:AR=1:2\).

3. By considering \(\overrightarrow{CM}\times\overrightarrow{CR}\) or otherwise, show that the points \(C\), \(M\) and \(R\) are collinear.[3]