Cambridge International Mathematics

Challenge

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12 Use vectors to find the remaining vertex of:

13 In the given figure, O is the origin and OUWV is a

parallelogram.

¡!

OU=u,

¡!

OV=v, and X lies on UW such that

UX : XW=4:1.

a Find, in terms ofuandv:

i

¡!

UX ii the position vector of X.

b OX is extended to Y so that

¡!

OY=^54

¡!

OX.

i Find

¡!

VY in terms ofuandv.

ii What can be deduced about V, W and Y? Explain your answer.

B,()3 -2

C,()2 -3

D,(-3 -5)

A

OU

V W

v

u

X

1aDraw any quadrilateral, not necessarily one of the special types. Accurately locate the

midpoints of the sides of the quadrilateral and join them to form another quadrilateral. What

do you suspect?

b Repeatawith a different quadrilateral two more times.

Make a detailed statement of what you suspect about the internal quadrilateral.

c Usingonly vector methods, prove that your suspicion inbis correct.

2a

b In the figure,

¡!

OA=a,

¡!

OB=b and BC is parallel

to OA and half its length.

i Explain why

¡!

OP could be written as k(b+^12 a)

wherekis a constant.

ii Explain why

¡!

OP could also be written as

a+t(b¡a) wheretis another constant.

iii Useato deduce the value ofkand hence write

¡!

OP

in terms of vectorsaandb.

O

A

P

B

C

a

b

504 Vectors (Chapter 24)

Supposeaandbare two non-zero vectors which are not parallel, and ra+sb=ma+nb

wherer,s,mandnare constants. Show that r=m and s=n.

IGCSE01

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Y:\HAESE\IGCSE01\IG01_24\504IGCSE01_24.CDR Thursday, 13 November 2008 9:09:16 AM PETER