Vectors
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 = (3) then the vector will look as follows:
           (2)

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 the scalar.

Example:
If a = ( 3 ), and b = 2a, sketch a and b.
         ( 2 )
If a = ( 3 ),   2a = ( 6 )
         ( 2 )           ( 4 )

Vector Manipulation

Example:
If a = (-5) and b = ( 2), find the magnitude of their resultant.
         ( 3)            ( 1)


The resultant of two or more vectors is their sum.
The resultant therefore is (-3).
                                      ( 4)
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.
Example: 3i + j  plus  5i - 4j =   8i - 3j. This is equivalent to:
( 3 ) + ( 5 )  =  ( 8 )
( 1 )    ( -4)      ( -3)

 © Matthew Pinkney