CAIE IGCSE Mathematics Vector Geometry Questions

Practise building vector routes, using section ratios and comparing scalar multiples to solve coordinate-free geometry and prove line relationships.

Exam points

follow a directed route through the diagram and simplify the result in the given base vectors
use a midpoint or section ratio to form the position vector of a point on a line segment
prove lines parallel or points collinear by exhibiting equal directions or a scalar multiple

Covered syllabus points

E7.4.1 Represent vectors by directed line segments.E7.4.2 Use position vectors.E7.4.3 Use the sum and difference of two or more vectors to express given E7.4.4 Use vectors to reason and to solve geometric problems

[Maximum number: 1]

F is the point (1,-4), $\overrightarrow{FG}=\binom{8}{-3}$ and $\overrightarrow{GH}=\binom{-12}{35}$ .
Find

the coordinates of the point G

[Maximum number: 5]

(a)

The position vector of P is $\begin{pmatrix}-2\\3\end{pmatrix}$ and the position vector of Q is $\begin{pmatrix}-2\\5\end{pmatrix}$ .

[ 2 ]

(i)

R is the point such that $\overrightarrow{P R}=3 \overrightarrow{P Q}$ .

Find the position vector of R.

()

[ 2 ]

$\begin{pmatrix}-2\\9\end{pmatrix}$

FT their $\overrightarrow{PQ}$
B1FT for $\begin{pmatrix}0\\6\end{pmatrix}$

(b)

Figure for Question (b) — CAIE IGCSE Mathematics Extended

$\overrightarrow{O T}=\mathbf{t}, \overrightarrow{O U}=\mathbf{u}$ and U Y=2 Y T.

[ 3 ]

(i)

Find $\overrightarrow{O Y}$ in terms of t and u.

Give your answer in its simplest form.

\overrightarrow{O Y}=

[ 2 ]

$\overrightarrow{OY}=\frac{2}{3}\mathbf{t}+\frac{1}{3}\mathbf{u}=\frac{1}{3}(2\mathbf{t}+\mathbf{u})$

M1 for $\overrightarrow{UY}=\frac{2}{3}(\mathbf{t}-\mathbf{u})$ oe or $\overrightarrow{TY}=\frac{1}{3}(\mathbf{u}-\mathbf{t})$ oe or correct route soi

(ii)

$\quad Z$ is on OT and YZ is parallel to UO.

Find $\overrightarrow{O Z}$ in terms of t and/or u.
Give your answer in its simplest form.

\overrightarrow{O Z}=

[ 1 ]

$\overrightarrow{OZ}=\frac{2}{3}\mathbf{t}$
cao

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Covered syllabus points

E7.4 Vector geometry question 1

E7.4 Vector geometry question 2

Question (a)

Question (i)

Question (b)

Question (i)

Question (ii)