As I said before, there are two ways of multiplying vectors together that make sense, the scalar product and the cross product. We'll postpone the discussion of cross products until a time closer to when we'll need them which will be in a moon or two from now.
The scalar product is also referred to as the dot product.
It takes two vectors, say 
and
and multiplies them together forming a scalar.
Scalars were defined above in section 4.1.1.
This is often notated as 
.
It is defined as follows:
where 
is the angle between the two vectors.
Notice that from the definition 
.
Also notice that this definition is independent of any
coordinate system that might get plonked down. The magnitudes
of , , and the angle between them are independent of
of any coordinate system. Since the answer doesn't depend on
coordinate system, the answer is a scalar (again see section
4.1.1).
We can also rewrite the dot product in a slightly different
way. Notice that 
is the component of
in the direction of . I can call this component 
in analogy to the notation 
which would be the component
of in the direction of the x axis. So this way we
can write
So what's 
? Well in the above equation set 
.
Then we have 
. But 
is
just the component of along the x axis which is . The
same is true for that other two components, in other words
Another neat thing about dot products is that they are distributive, that is
This can be seen as follows. From what was said in the
paragraph above, 
,
but 
. So we have
.
Now we can use the above to evaluate in
Cartesian coordinates:
Now 
, because
the angle between a vector and itself is zero. Also
because these
three vectors are perpendicular, so the cos term is zero.
Using this fact plus the fact the dot product is distributive
you can just multiply out the equation above. Most of the terms
are zero, and you end up getting
This is a very useful formula. Let's look at a useful illustration of it
right now.