Cambridge International Mathematics

Notice that a=

μ

6

3

¶

and b=

μ

2

1

¶

are such that a=3b.

We can see that akb.

Notice also that jaj=

p

36 + 9

=

p

45

=3

p

5

=3jbj:

Consider the vector ka which is parallel toa.

² If k> 0 then ka has the same direction asa.

² If k< 0 then ka has the opposite direction toa.

²jkaj=jkjjaj i.e., the length ofkais themodulusofktimes the length ofa.

If two vectors are parallel and have a point in common then all points on the vectors are collinear.

Example 15 Self Tutor

What two facts can be deduced aboutpandqif:

a p=5q b q=¡^34 p?

a p=5q

) pis parallel toq and jpj=j 5 jjqj=5jqj

) pis 5 times longer thanq, and they have the same direction.

b q=¡^34 p

) qis parallel top and jqj=

̄

̄¡^3

4

̄

̄jpj=^3

4 jpj

) qis^34 as long asp, but has the opposite direction.

EXERCISE 24G

1 What two facts can be deduced if:

a p=2q b p=^12 q c p=¡ 3 q d p=¡^13 q?

2

μ

5

2

¶

and

μ

k

¡ 4

¶

are parallel. Findk.

3 Use vector methods only to show that P(¡ 2 ,5),Q(3,1),R(2,¡1)

and S(¡ 3 ,3), form the vertices of a parallelogram.

4 Use vector methods to find the remaining vertex of parallelogram ABCD:

ab

a

b

D,()-1 ¡1

C

B,()5 ¡7

A,()2 ¡3

Vertices are always

listed in order, so

PQRS is

P

Q

R

S

P

Q

R

S

either

or

A,()4 ¡3

D

C,()7 -2

B,()2 ¡1

498 Vectors (Chapter 24)

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Y:\HAESE\IGCSE01\IG01_24\498IGCSE01_24.CDR Monday, 27 October 2008 2:27:13 PM PETER