Sketching the information given will help you to visualise the problem.
PN = 2NQ can be described as "N is between P and Q and PN is twice as long as NQ".

cie-igcse-2018-may-jun-tz2-p2-q25b

We can see that N is two thirds of the way from P to Q.

Write the coordinates as position vectors.

$\vec{O P} = (\begin{matrix} 16 \\ 9 \end{matrix}) \vec{O Q} = (\begin{matrix} 22 \\ 24 \end{matrix})$

Find the vector from P to Q using $\vec{P Q} = \vec{O Q} - \vec{O P}$ .

$\begin{array}{rcl} \vec{P Q} & = & (\begin{matrix} 22 \\ 24 \end{matrix}) - (\begin{matrix} 16 \\ 9 \end{matrix}) = (\begin{matrix} 6 \\ 15 \end{matrix}) \end{array}$

Multiply this by two thirds.

$\begin{array}{rcl} \vec{P N} & = & \frac{2}{3} \vec{P Q} = \frac{2}{3} \times (\begin{matrix} 6 \\ 15 \end{matrix}) = (\begin{matrix} 4 \\ 10 \end{matrix}) \end{array}$

[1]

Add this on to the position vector of P to get the position vector of N using $\vec{O N} = \vec{O P} + \vec{P N}$ .

$\begin{array}{rcl} O N & = & (\begin{matrix} 16 \\ 9 \end{matrix}) + (\begin{matrix} 4 \\ 10 \end{matrix}) = (\begin{matrix} 20 \\ 19 \end{matrix}) \end{array}$

Rewrite the position vector as coordinates.

$N = (20, 19)$ [1]

Vectors (CIE IGCSE Maths: Extended)

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