2019 ASRJC Promo Q7

Referred to the origin \(O\), \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) are non-zero and non-parallel vectors denoting the position vectors of the points \(A\), \(B\)and \(C\) respectively.

1. Given that \(\mathbf{a}\times \mathbf{b}=3\mathbf{a}\times \mathbf{c}\), show that \(\mathbf{b}-3\mathbf{c}=\lambda \mathbf{a}\) where \(\lambda \) is a scalar.[2]

The point \(M\) is the mid-point of \(OC\)and the point \(N\) lies on \(OB\) produced such that \(3ON=5OB\). The point \(P\) lies on \(MN\) such that \(MP:MN=2:3\).

2. Show that position vector of \(P\) is \(\frac{10}{9}\mathbf{b}+\frac{1}{6}\mathbf{c}\).[1]

It is given that \(\mathbf{b}\) is a unit vector, \(\left| \mathbf{c} \right|=\sqrt{2}\), \(\left| \mathbf{b}-3\mathbf{c} \right|=5\) and the angle between \(\mathbf{b}\) and \(\mathbf{c}\) is \(45{}^\circ \).

3. Find the exact length of projection of \(\overrightarrow{OP}\) on \(\overrightarrow{OA}\).[4]