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In the first two sections of this chapter we looked at
vectors in 2-space and 3-space. You
probably noticed that with the exception of the cross product (which is only
defined in 3-space) all of the formulas that we had for vectors in 3-space were
natural extensions of the 2-space formulas.
In this section we’re going to extend things out to a much more general
setting. We won’t be able to visualize
things in a geometric setting as we did in the previous two sections but things
will extend out nicely. In fact, that
was why we started in 2-space and 3-space.
We wanted to start out in a setting where we could visualize some of
what was going on before we generalized things into a setting where
visualization was a very difficult thing to do.
So, let’s get things started off with the following
definition.
In the previous sections we were looking at 
(what we were calling 2-space) and (what we were calling 3-space). Also the more standard terms for 2-tuples and
3-tuples are ordered pair and ordered triplet and that’s the terms
we’ll be using from this point on.
Also, as we pointed out in the previous sections an ordered
pair, ,
or an ordered triplet, ,
can be thought of as either a point or a vector in or respectively.
In general an ordered n-tuple,
,
can also be thought of as a “point” or a vector in . Again, we can’t really visualize a point or a
vector in ,
but we will think of them as points or vectors in anyway and try not to worry too much about the
fact that we can’t really visualize them.
Next, we need to get the standard arithmetic definitions out
of the way and all of these are going to be natural extensions of the
arithmetic we saw in and .
The basic properties of arithmetic are still valid in so let’s also give those so that we can say
that we’ve done that.
The proof of all of these come directly from the definitions
above and so won’t be given here.
We now need to extend the dot product we saw in the previous
section to and we’ll be giving it a new name as well.
So, we can see that it’s the same notation and is a natural
extension to the dot product that we looked at in the previous section, we’re
just going to call it something different now.
In fact, this is probably the more correct name for it and we should
instead say that we’ve renamed this to the dot product when we were working
exclusively in and .
Note that when we add in addition, scalar multiplication and
the Euclidean inner product to we will often call this Euclidean n-space.
We also have natural extensions of the properties of the dot
product that we saw in the previous section.