2017 MJC J1 MYE Q5

With respect to the origin \(O\), the position vectors of points \(A\), \(B\) and \(C\) are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}=\lambda \mathbf{a}+\mu \mathbf{b}\) where \(\lambda \), \(u>0\) respectively. Given that the angle \(AOB\) is acute, show that the length of projection of \(\mathbf{c}\) on \(\mathbf{b}\) can be expressed as \(\frac{\lambda \mathbf{a}\cdot \mathbf{b}}{\left| \mathbf{b} \right|}+\mu \left| \mathbf{b} \right|\).

Given further that \(OACB\) is a unit rhombus and angle \(AOB\) is \({{45}^{\circ }}\), deduce the exact length of projection of \(\mathbf{c}\) on \(\mathbf{b}\).[5]

With respect to the origin \(O\), the position vectors of non-collinear points \(P\), \(Q\) and \(R\) are \(\mathbf{p}\), \(\mathbf{q}\) and \(\mathbf{r}\) respectively. Show that the area of triangle \(PQR\) is \(\frac{\left| \mathbf{p}\times \mathbf{q}+\mathbf{q}\times \mathbf{r}+\mathbf{r}\times \mathbf{p} \right|}{2}\).

Hence find an expression for the perpendicular distance from \(P\) to the line passing through points \(Q\) and \(R\) in terms of \(\mathbf{p}\), \(\mathbf{q}\) and \(\mathbf{r}\).[4]