Finding the position vector of extensions

Question

I have this triangle

And this problem

OM is extended to C. The position vector of C is a+kb. Find k

How can i find k here? I know the position vector of M is '2/5a+3/5b'.

Answer 1

Observe that \begin{align} \vec{OM}=t\mathbf{a}+(1-t)\mathbf{b} \end{align} for some $0\leq t \leq 1$. Here, it is clear that $t=\frac{2}{5}$. Hence we have \begin{align} \vec{OM}=\frac{2}{5}\mathbf{a}+\frac{3}{5}\mathbf{b} \ \implies \ \ \frac{5}{2}\vec{OM} = \mathbf{a}+\frac{3}{2}\mathbf{b}. \end{align}