CAIE IGCSE Mathematics Extended E7.4 Vector…metry

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: 6]

The diagram shows triangle OPT.

Figure for Question 2 — 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.

(a)

Find, in terms of t and p, in its simplest form

[ 4 ]

(i)

$\overrightarrow{P L}$

[ 2 ]

$-\frac13\mathbf{p}+\frac13\mathbf{t}$ or $\frac13(-\mathbf{p}+\mathbf{t})$ final answer

M1 for $\overrightarrow{PT}=-\mathbf{p}+\mathbf{t}$ or $\overrightarrow{TP}=-\mathbf{t}+\mathbf{p}$ .

(ii)

$\overrightarrow{K L}$ .

[ 2 ]

$\frac23\mathbf{p}-\frac13\mathbf{t}$ or $\frac13(2\mathbf{p}-\mathbf{t})$ final answer

M1 for a correct route.

(b)

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.

[ 2 ]

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

[Maximum number: 5]

NOT TO SCALE

ABCD is a parallelogram.
$\overrightarrow{A B}=\mathbf{m}$ and $\overrightarrow{A D}=\mathbf{p}$ .
F is a point on BC and B F=4 F C.
E is a point on AD and A E: E D=1: 2.

(a)

Find $\overrightarrow{EF}$ , in terms of $\mathbf{m}$ and $\mathbf{p}$ , in its simplest form.

[ 3 ]

$\mathbf{m}+\frac{7}{15} \mathbf{p}$

M2 for correct unsimplified expression in terms of $\mathbf{m}$ and $\mathbf{p}$

or M1 for a correct route

(b)

EF and DC are extended to meet at the point G.

Find $\overrightarrow{CG}$ , in terms of $\mathbf{m}$ and/or $\mathbf{p}$ , in its simplest form.

[ 2 ]

$\frac{3}{7} \mathbf{m}$

M1 for using the ratio $\frac{2}{3}[\mathbf{p}]: \frac{1}{5}[\mathbf{p}]$

[Maximum number: 3]

NOT TO SCALE

ABCD is a parallelogram.
$\overrightarrow{A B}=\mathbf{m}$ and $\overrightarrow{A D}=\mathbf{p}$ .
F is a point on BC and B F=4 F C.
E is a point on AD and A E: E D=1: 2.

Find $\overrightarrow{EF}$ , in terms of $\mathbf{m}$ and $\mathbf{p}$ , in its simplest form.

$\mathbf{m}+\frac{7}{15} \mathbf{p}$

M2 for correct unsimplified expression in terms of $\mathbf{m}$ and $\mathbf{p}$

or M1 for a correct route

[Maximum number: 2]

The diagram shows triangle OPT.

Figure for Question 5 — 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 Vector Geometry Topic Practice

Covered syllabus points

Question 1

Question 2

Question (a)

Question (i)

Question (ii)

Question (b)

Question 3

Question (a)

Question (b)

Question 4

Question 5