Cambridge Additional Mathematics

294 Vectors (Chapter 11)

When we apply vectors to problems in the real world, we

often consider the combined effect when vectors are added

together. This sum is called theresultant vector.

The diagram shows an example of vector addition. Two tug

boats are being used to pull a ship into port. If the tugs tow

with forcesF 1 andF 2 then the resultant force isF 1 +F 2.

Example 12 Self Tutor

In still water, Jacques can swim at 1 : 5 ms¡^1. Jacques is

at point A on the edge of a canal, and considers point B

directly opposite. A current is flowing from the left at a

constant speed of 0 : 5 ms¡^1.

a If Jacques dives in straight towards B, and swims

without allowing for the current, what will his actual

speed and direction be?

b Jacques wants to swim directly across the canal to point B.

i At what angle should Jacquesaimto swim in order that the current will correct his direction?

ii What will Jacques’ actual speed be?

Suppose cis the current’s velocity vector,

sis the velocity vector Jacques would have if the water was still, and

f=c+s is Jacques’ resultant velocity vector.

a Jacques aims directly across the river, but the current takes him downstream to the right.

jfj^2 =jcj^2 +jsj^2

=0: 52 +1: 52

=2: 5

) jfj¼ 1 : 58

tanμ=

0 : 5

1 : 5

) μ¼ 18 : 4 ±

Jacques has an actual speed of approximately 1 : 58 ms¡^1 and his direction of motion is

approximately 18 : 4 ±to the right of his intended line.

b Jacques needs to aim to the left of B so the current will correct his direction.

i sinÁ=

0 : 5

1 : 5

) Á¼ 19 : 5 ±

Jacques needs to aim approximately 19 : 5 ±to the left of B.

F Problems involving vector operations

A

B

current

05 .ms-1

B

s

c

f

A

q

current

05. ms-1

F 1 F^2

FF 12 +

F 2

cyan magenta yellow black

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_11\294CamAdd_11.cdr Monday, 6 January 2014 1:04:59 PM BRIAN