A vector quantity has both length (magnitude) and direction. The opposite is a scalar quantity, which only has magnitude. Vectors can be denoted by AB, a, or AB (with an arrow above the letters).
If a = then the vector will look as follows:

NB1: When writing vectors as one number above another in brackets, this is known as a column vector.
NB2: In textbooks and here, vectors are indicated by bold type. However, when you write them, you need to put a line underneath the vector to indicate it.

Multiplication by a Scalar

When multiplying a vector by a scalar (i.e. a number), multiply each component of the vector by that number.

Example

If a = , and b = 2a, sketch a and b.        

If a = , then  2a =

Vector Manipulation

Example

If a = and b = , find the magnitude of their resultant.

The resultant of two or more vectors is their sum.
The resultant therefore is .
The magnitude of this is Ö(-3² + 4²) = Ö(9 + 16) = Ö(25) = 5

The addition and subtraction of vectors can be shown diagrammatically. To find a + b, draw a and then draw b at the end of a. The resultant is the line between the start of a and the end of b.
To find a - b, find -b (see above) and add this to a.

Example

Unit Vectors

A unit vector has a magnitude of 1. The unit vector in the direction of the x-axis is i and the unit vector in the direction of the y-axis is j. For example on a graph, 3i + 4j would be at (3 , 4). This method is another method of writing down vectors. It also makes adding and subtracting vectors easy: you just add the i terms together and add the j terms together.
For example: 3i + j  plus  5i - 4j =   8i - 3j. 

This is the same as writing is as:

Copyright © Matthew Pinkney 2003