From now on I will denote points in the plane by lower case boldface letters,
e.g.
. If I specify a point
and do not explicitly write
down its
components, you should assume
, etc. The one exception to this rule is that I will always
take

If , we will write for ; i.e., . We will abbreviate by , and we will write .

We have the following law that resembles the associative law for
multiplication:

We have the following distributive laws:

Also,

All of these properties follow easily from the corresponding properties of real numbers. I will prove the commutative law and one of the distributive laws, and omit the remaining proofs.

Proof of Commutative Law: Let
be points in
.
By
the commutative law for
,

Hence

and hence .

Proof of (4.3): Let
and let
. By
the
distributive law for
we have

Hence,

i.e,

**Remark**: Let
,
be points in
such that
and
are not all in a straight line. Then
is the vertex opposite
in the parallelogram whose other three vertices are
and
.

Proof: In this proof I will suppose and , so that neither of is a vertical line. (I leave the other cases to you.) The slope of line is , and the slope of is . Thus the lines and are parallel.

The slope of line
is
,
and
the slope of
is

. Thus the lines
and
are parallel. It follows that the figure
is a
parallelogram,
i.e.,
is the fourth vertex of a parallelogram having
and
as its other vertices.

In figure b), and are the vertices of a regular hexagon with . Sketch the points , and as accurately as you can. (This problem should be done geometrically. Do not calculate the coordinates of any of these points.)

Hence a point is in if and only if there is a number such that

Now

Thus

If then

Now as runs through all values in , we see that also takes on all values in so we get the same description for when as we do when . Note that this description is exactly what you would expect from the pictures, and that it also works for vertical segments.