Cambridge International AS and A Level Mathematics Pure Mathematics 1

Vectors in three dimensions

Vectors in three dimensions

P1^

8

Equal vectors

The statement that two vectors a and b are equal means two things.

●●The direction of a is the same as the direction of b.

●●The magnitude of a is the same as the magnitude of b.

If the vectors are given in component form, each component of a equals the
corresponding component of b.

Position vectors

Saying the vector a is given by 3 i + 4 j + k tells you the components of the vector,
or equivalently its magnitude and direction. It does not tell you where the vector
is situated; indeed it could be anywhere.

All of the lines in figure 8.9 represent the vector a.

There is, however, one special case which is an exception to the rule, that of a
vector which starts at the origin. This is called a position vector. Thus the line

joining the origin to the point P(3, 4, 1) is the position vector

3

4

1

















or  3 i +   4 j + k.

Another way of expressing this is to say that the point P(3, 4, 1) has the position

vector

3

4

1

















.

k

i j

a

a

a

a

Figure 8.9