Worked Examples in Science and Mathematics

What do we mean when we use the expression "vector"?

A vector, in science and mathematics is an entity with both magnitude and direction. Related to the vector is the property known as a "scalar" which has magnitude only. In this section will concentrate on the vectors and how they can be manipulated.

As the definition would suggest, vectors can be used to determine the relative positions of bodies with respect to each other, and in fact we use this quite regularly when looking at coordinates in 2 and 3 dimensions.

When an object moves 1 point another, it usually does so in a straight line. This movement, as well as being a direct "shortest distance from point to point" can also be regarded as the displacement occurring in each of the three-dimensions.

Take a look at the diagram:

In three-dimensional space, we start from a point labelled as 0, 0, 0 referred to as the origin, we can move in any direction as shown by the red arrow. If we label our three-dimensional coordinates from the origin point in the X, Y and Z directions we can look at the relative components of the movement in the direction of the arrow according to the proportion of displacement along each of the 3 axes.

In this section we will look at two-dimensional vectors only, so will disregard the 3rd (Z) axis. A vector therefore can be shown a straight line from one point to another, the direction of the vector should be quite clear and the magnitude of the vector will be indicated by the length of the drawn line.

If we take a look at the vector represented by the green arrow OP we can describe this as a vector with a magnitude of the length of the line in the direction from O to P with an X component of 8 units and a white component of 5 units. It will be a simple case of using Pythagoras theorem to evaluate the magnitude of the vector itself. Vector is usually written as the letters representing the vector with an arrow above, or as a column vector with the X component above the Y component.

Selecting anyone of the 4 examples above, the red one for example we can see that the vector has a magnitude shown by the length of its line and extends 2 units in the positive horizontal direction (X) and 4 units in the positive vertical direction (Y).