Cambridge Additional Mathematics

Vectors (Chapter 11) 281

Now consider the vector from point A to point B. We say that:

²

¡!

AB is the vector whichoriginatesat A and

terminatesat B

²

¡!

AB is theposition vectorof B relative to A.

When we plot points in the Cartesian plane, we move first in thex-direction and then in they-direction.

For example, to plot the point P(2, 5), we start at the

origin, move 2 units in thex-direction, and then 5 units in the

y-direction.

We therefore say that the vector from O to P is

¡!

OP=

μ

2

5

¶

.

Suppose that i=

μ

1

0

¶

is a vector of length 1 unit in the positivex-direction

and that j=

μ

0

1

¶

is a vector of length 1 unit in the positivey-direction.

¡!

OP=2i+5j

)

μ

2

5

¶

=2

μ

1

0

¶

+5

μ

0

1

¶

The point P(x,y) hasposition vector

¡!

OP=

μ

x

y

¶

=xi+yj.

unit vector form

i=

μ

1

0

¶

is thebase unit vectorin thex-direction.

j=

μ

0

1

¶

is thebase unit vectorin they-direction.

The set of vectors fi,jg is thestandard basisfor the 2 -dimensional(x,y) coordinate system.

A

B

component form

y

x

5

2

P,(2 5)

O

iand are calledj unit vectors

because they have length. 1

y

x

P,(2 5)

i i

j

j

j

j

j

O

¡! 2 _____

OP={}5__________ _

We can see that moving from O to P is equivalent to 2 lots ofi

plus 5 lots ofj.

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