Vectors


Introduction


Whereas numbers just have a magnitude, vectors have both a magnitude and direction. The operations of addition and subtraction of vectors will need to take this direction into account as well as the magnitude.

A vector is usually represented by an arrow pointing in the appropriate direction, the length of the arrow representing the magnitude of the vector. In print, they are often named using boldface to distinguish them from 'ordinary numbers', for example

a,    b

In hand-writing, it is usual to write the letter and underline it.

The vectors dealt with at GCSE level are 'free vectors', that means they are not actually attached to any point. If I translate a vector d to another position it still remains d.

So, likewise, if two vectors a and b point in the same direction (i.e. are parallel) and are of the same length (or magnitude), then the two vectors are identical - they do not have to be represented as coincident on the page.

Addition

The diagram shows the construction of a parallogram to add the vectors A and B - the addition of these two vectors produces C.

The vectors to added are specifically the two at the bottom left. The other two sides of the parallelogram have been labelled as A and B respectively. This follows on from the ideas expressed in the 'Introduction' - namely

  • A vector can be translated and still retain its identity

    or

  • Two vectors parallel and of equal length are identical.

We can see that an alternative way of adding vectors presents itself - 'line the vectors up' with the second vector attached to the head of the first vector. The required addition is represented by a line drawn from the end of the first vector to the tip of the second vector.

Diagram

This can be generalized to adding any number of vectors

Diagram

Tug

Subtraction