2026 HCI Promo Q9

\(A\), \(B\) and \(C\) are non-collinear points in the three-dimensional space relative to the origin \(O\) with position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively. Suppose \(\mathbf{a}=|\mathbf{b}|\mathbf{c}+|\mathbf{c}|\mathbf{b}\).

1. Show that \(\cos\angle AOB=\dfrac{(\mathbf{b}\cdot\mathbf{c})+|\mathbf{b}||\mathbf{c}|}{|\mathbf{a}|}\) and conclude that the angle between \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) is equal to the angle between \(\overrightarrow{OA}\) and \(\overrightarrow{OC}\).[4]
2. Show that the area of the triangle \(AOB\) can be expressed in the form \(k|\mathbf{b}\times\mathbf{c}|\), where \(k\) is a constant in terms of \(|\mathbf{b}|\) to be determined.[3]
3. It is now given that \(\mathbf{b}\) and \(\mathbf{c}\) are unit vectors and \(\angle AOB=15^\circ\). Find the area of \(OBAC\).[2]