Cambridge Additional Mathematics

Vectors (Chapter 11) 293

Example 11 Self Tutor

If a=3i¡j, find:

a a unit vector in the direction ofa

b a vector of length 4 units in the direction ofa

c vectors of length 4 units which are parallel toa.

a jaj=

p

32 +(¡1)^2

=

p

9+1

=

p

10 units

) the unit vector is p^110 (3i¡j)

=p^310 i¡p^110 j

b This vector is p^410 (3i¡j)

=p^1210 i¡p^410 j

c The vectors are p^1210 i¡p^410 j and ¡p^1210 i+p^410 j.

EXERCISE 11E

1 Findrgiven that a=

μ

2

¡ 1

¶

and b=

μ

¡ 6

r

¶

are parallel.

2 Findagiven that

μ

3

¡ 1

¶

and

μ

a

2

¶

are parallel.

3 What can be deduced from the following?

a

¡!

AB=3

¡!

CD b

¡!

RS=¡^12

¡!

KL c

¡!

AB=2

¡!

BC

4 If a=

μ

2

4

¶

, write down the vector:

a in the same direction asaand twice its length

b in the opposite direction toaand half its length.

5 Find the unit vector in the direction of:

a i+2j b i¡ 3 j c 2 i¡j

6 Find a vectorvwhich has:

a the same direction as

μ

2

¡ 1

¶

and length 3 units

b the opposite direction to

μ

¡ 1

¡ 4

¶

and length 2 units.

7 Ais(3,2) and point B is 4 units from A in the direction

μ

1

¡ 1

¶

.

a Find

¡!

AB. b Find

¡!

OB using

¡!

OB=

¡!

OA+

¡!

AB.

c Hence deduce the coordinates of B.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_11\293CamAdd_11.cdr Monday, 6 January 2014 9:56:19 AM BRIAN