Vectors can be used to describe lines in computer graphics.  To use a simple
example, suppose that a line begins at the origin of a graph ( 0, 0 ) and its end
point is x= 2, y = 4.  The line from ( 0, 0 ) to ( 2, 4 ) is a vector.  A line
beginning from ( 100, 100 ) and going to ( 102, 104 ) is the exact same vector
since in vector mathematics all vectors with the exact same length are
considered to be the same vector. Normally vectors are manipulated using two
mathematical operations, addition and scalar mulitiplication.  

Vectors can be written as either row or column vectors.  Column vectors are
written horizontally.  For example the vectors described above would be
written as :

   (  2  )
      4 

and written as a row vector : 

  (2, 4) . 

Let's assume that we wish to add two vectors (2, 4) and (3, 2).  (Vectors can
only be added when they are in the same vector space - in this example R2). 
The components of the vector are added to each other to produce the final
result (so then the x components are added together and the y components are
added together) :

(x1, y1) + (x2 + y2) =   (x1 + x2, y1 + y2) = (2 + 3, 4 + 2) = (6, 5)

Vectors are usually given variable names and thus we could write :

u = (2, 4) and v = (3, 2) so then
u + v = (u1 + v1, u2 + v2) = (6, 5)


Scalar multiplication results in a new (longer or shorter vector line segment),
and is accomplished by multiplying the scalar with the components of the
vector.  For example if we have a vector u = (2, 4) and we multiply by the
scalar k = 4 we get

k * u = (k * u1, k * u2) = (4 * 2, 4 * 4) = (8, 16)

If k is greater than 0 the direction of the vector remains unchanged and only its
magnitude (or length) changes while if k is negative the vector direction is the
opposite to that which it previously had.  (On a graph the angle of the line
remains unchanged, while only the length of the line changes when applying
scalar mathematics, so intuitively it seems right to assume that scalar
mathematics could be used to scale illustrations to various sizes, provided that
the graphic is defined using vectors).

Vectors can also be subtracted.  For example if u = (4, 2) and v = (2, 3) then 

u - v = (4 - 2, 2 - 3) = (2, -1)

This is also defined as the difference between the two vectors.  So then
difference between the first vector us and the second vector v is (2, -1) since 4
is 2 greater than 2 and 2 is one less than 3.

Vector mathematics can also be used to solve equations of the type 2x + 3y +
4z, assuming that x, y, and z are vectors.  First the scalar multiplication is
performed and then the vector components are added together.

It is also possible to solve equations where there is an unknown variable.  For
example if u and v are vectors and u = v then the components in both u and v
are identical.  So then if u = (2, 4) and u = v then v = (2, 4).  So if u = v and u
= (2, 4) and v = ( x, 4) then we know that x = 2.  Linear equations can be used
to solve for the unknowns in vector mathematics.  For example if u = v and u =
(6, y) and v = b(2, 4) then 

(6, y) = (2a, 4a) so, since equal vectors are component wise equal then 
6 = 2a and y = 4a   therefore a = 6 / 2 so a = 3 which means that y = 4 * 3 =
12.
The solution then is that u = (6, 12) and v = 3 * (2, 4) = (6, 12)