10.
Figure 7
Figure 7 shows a sketch of triangle \(O A B\).
The point \(C\) is such that \(\overrightarrow { O C } = 2 \overrightarrow { O A }\).
The point \(M\) is the midpoint of \(A B\).
The straight line through \(C\) and \(M\) cuts \(O B\) at the point \(N\).
Given \(\overrightarrow { O A } = \mathbf { a }\) and \(\overrightarrow { O B } = \mathbf { b }\)
- Find \(\overrightarrow { C M }\) in terms of \(\mathbf { a }\) and \(\mathbf { b }\)
- Show that \(\overrightarrow { O N } = \left( 2 - \frac { 3 } { 2 } \lambda \right) \mathbf { a } + \frac { 1 } { 2 } \lambda \mathbf { b }\), where \(\lambda\) is a scalar constant.
- Hence prove that \(O N : N B = 2 : 1\)