INTRODUCTION TO VECTORS

1. Scalars and vectors

A SCALAR is a quantity which has MAGNITUDE (numerical size) only. Examples of scalars are the natural numbers, speed, distance, energy, charge, volume and temperature.

Scalar quantities can be manipulated by the laws of arithmetic applicable to natural numbers. Many physical quantities can be added together in the same way as natural numbers. For example, if we first put 100 cm3 of water into a cup and then put in an additional 150 cm3, the cup will contain 250 cm3 of water.

Similarly, if you were to run around a square field having a side length 100 m, you would have run a total distance of 400 m.

Such quantities can also be subtracted in the usual way. For example, if you were to eat 100 g cheese from a piece of mass 500 g, the mass of the remaining piece would be 400 g.

We have used volume, distance and mass as examples of physical quantities called SCALARS. Other examples are time, temperature and any natural number. The VALUE of a scalar is called its MAGNITUDE

A VECTOR is a quantity which has both a MAGNITUDE AND A DIRECTION. Vectors arise naturally as physical quantities. Examples of vectors are displacement, velocity, acceleration, force and electric field.

Special arithmetic rules must be obeyed when adding vectors together. Much of this topic is devoted to these rules!

Some physical quantities cannot be added in the simple way described for scalars.
For example, if you were to walk 4 m in a northerly direction and then 3 m in an easterly direction, how far would you be from your starting point? The answer is clearly NOT 7 m! To find the answer, one could draw a scale diagram (1 cm = 1 m) such as is shown on the right:

One could also calculate the distance from the starting point using the theorem of Pythagoras, i.e.

It is also useful to know in which direction one has moved from the starting point. This can also be measured from the diagram or calculated from simple trigonometry:

You could have reached the same final position by walking 5 m in the direction 36.9 east of north. This is the result of adding "4 m north" and "3 m east". The physical quantities, 4 m north, 3 m east and 5 m 36.9 east of north require both a magnitude and a direction to fully describe them. These quantities are called DISPLACEMENTS. Displacement is an example of a VECTOR QUANTITY.

Other examples of vector quantities that you will encounter are velocity, acceleration and force. ALL VECTOR QUANTITIES CAN BE ADDED TOGETHER IN THE SAME WAY AS DISPLACEMENTS.


2. Vector notation

Vectors are distinguished from scalars by writing them in special ways. A widely used convention is to denote a vector quantity in bold type, such as A. In some books, you may also encounter the notation or . The South African Department of Basic Education, in their examination papers and memoranda, does not use any special notation differentiating between vectors and scalars, and that is the style that will be used here.

The magnitude of a vector A is written as |A|.

3. Vector representation

Since several important physical quantities are vectors, it is useful to agree on a way for representing them and adding them together.

In the example involving displacement, we used a scale diagram in which displacements were represented by arrows which were proportionately scaled and orientated correctly with respect to our axes (i.e., the points of the compass). This representation can be used for all vector quantities provided the following rules are followed:

  1. The reference direction is indicated.
  2. The scale is indicated.
  3. The vectors are represented as arrows with a length proportional to their magnitude and are correctly orientated with respect to the reference direction.
  4. The direction of the vector is indicated by an arrowhead.
  5. The arrows should be labelled to show which vectors they represent.

For example, the diagram above on the right shows two vectors A and B, where A has a magnitude of 3 units in a direction parallel to the reference direction and B has a magnitude of 2 units and a direction 60 clockwise to the reference direction:

4. Equality of vectors


Two vectors are equal when they have the same magnitude and direction, irrespective of their point of origin. In the diagram on the right, A = B, since they have the same magnitude and direction.


5. Negative vectors

A vector having the same magnitude but opposite direction to a vector A, is -A.


6. Additional questions