Cambridge International AS and A Level Mathematics Pure Mathematics 1

Vectors

P1^

8

ExamPlE 8.5 Points L, M and N have co-ordinates (4, 3), (−2, −1) and (2, 2).

(i) Write down, in component form, the position vector of L and the vector M

→

N.

(ii)    What do your answers to part (i) tell you about the lines OL and MN?

SOlUTION

(i) The position vector of L is O

→

L = 4

3









.

    The vector M

→

N is also 4

3









(see figure 8.10).

(ii) Since O

→

L = M

→

N, lines OL and MN are parallel and equal in length.

Note

A line joining two points, like MN in figure 8.10, is often called a line segment,

meaning that it is just that particular part of the infinite straight line that passes

through those two points.

The vector M

→

N is an example of a displacement vector. Its length represents the

magnitude of the displacement when you move from M to N.

The length of a vector

In two dimensions, the use of Pythagoras’ theorem leads to the result that a

vector a 1 i + a 2 j has length | a | given by

| a | = aa 12 +^22.

y

4

3

2

1

–1

–2 –1 O 1 2 3 4 x

M

N

L

Figure 8.10