Vectors aren't really necessary to understand physics but they're
really cool and simplify understanding of a lot of problems
enormously. Without vectors mathematical descriptions are much
more cumbersome. A vector is something with both a magnitude
and a direction. It is often thought of as an arrow like so:
. The length of the arrow is its magnitude, and
obviously the direction that its pointing is the direction (duh).
You can define operations on vectors analogous to
the addition of regular (real) numbers. You can add and subtract
vectors, and there are two common ways of multiplying them
together.
Vectors are most commonly notated in books by using bold face.
A vector named ``A" would be notated 
to distinguish
it from a regular real number. When writing vectors, it
is common to represent it by placing a little arrow right
over the top of the letter (e.g. 
).
As I just said, it's nice to think about a vector as being
an arrow, having a magnitude (the length of the arrow) and
a direction. Two vectors (e.g. 
and )
that have the same length and go in the same direction are equal,
even if they don't start off at the same point in space,
as with the two arrows above. However they are different
if either their magnitudes or directions differ (e.g.
and .
From a mathematical point of view, its good to represent vectors by real numbers and that's what we'll talk about now.