CAIE IGCSE Mathematics Extended E7.4.4 Use Vectors to Reason and to Solve Geometric Problems Questions

Practise Extended CAIE IGCSE Mathematics by converting ratios into position vectors, comparing scalar multiples to prove parallel lines and establishing collinearity with a shared direction.

Exam points

Calculate a point’s position vector from a stated ratio along a segment or an extended line.
Show two lines are parallel by deriving one displacement vector as a scalar multiple of the other.
Prove three points are collinear by giving a common direction vector and the required shared point.

[Maximum number: 2]

The diagram shows triangle OPT.

Figure for Question CAIE IGCSE Mathematics Extended E7.4.4 Use Vectors to Reason and to Solve Geometric Problems Questions question 1 — CAIE IGCSE Mathematics Extended

NOT TO
SCALE

In the diagram $\overrightarrow{O T}=\mathbf{t}$ and $\overrightarrow{O P}=\mathbf{p}$ .
O K: K T=2: 1 and T L: L P=2: 1.

KL is extended to the point M.
$\overrightarrow{K M}=-\frac{2}{3} \mathbf{t}+\frac{4}{3} \mathbf{p}$ .

Show that M lies on OP extended.

$\overrightarrow{OM}=\frac23\mathbf{t}-\frac23\mathbf{t}+\frac43\mathbf{p}$ oe
$\overrightarrow{OM}=\frac43\mathbf{p}$ and $\overrightarrow{OM}$ is a multiple of $\overrightarrow{OP}$ oe

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CAIE IGCSE Mathematics Extended E7.4.4 Use Vectors to Reason and to Solve Geometric Problems Questions question 1