Method 1
We need to identify a route from $O$ to $T$ in terms of $p$ and $q$ . A possible route is $O$ to $Q$ to $T$ , or $\overset{⇀}{O T} = \overset{⇀}{O Q} + \overset{⇀}{Q T}$ .
If $\overset{⇀}{Q T} : \overset{⇀}{T P} = 2 : 1$ then $\overset{⇀}{Q T} = \frac{2}{3} \overset{⇀}{Q P}$ so $\overset{⇀}{O T} = \overset{⇀}{O Q} + \frac{2}{3} \overset{⇀}{Q P}$

$\begin{array}{rcl} \overset{⇀}{Q P} & = & - q + p \\ \frac{2}{3} \overset{⇀}{Q P} & = & \frac{2}{3} (- q + p) \end{array}$

[1]

Therefore

$\begin{array}{rcl} \overset{⇀}{O T} & = & \overset{⇀}{O Q} + \frac{2}{3} \overset{⇀}{Q P} \\ = & q + \frac{2}{3} (- q + p) \end{array}$

Expand the bracket and collect like terms

$\begin{array}{rcl} = & q - \frac{2}{3} q + \frac{2}{3} p \\ = & \frac{2}{3} p + \frac{1}{3} q \end{array}$

$\begin{array}{rcl} \overset{⇀}{O T} & = & \frac{2}{3} p + \frac{1}{3} q \end{array}$ [1]

Vectors (Cambridge O Level Maths)

Exam Questions