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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

4.1   Definition (Addition of Points) If and are points in and , we define

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

4.2   Theorem. Let be arbitrary points in and let be arbitrary numbers. Then we have:

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

We have the following distributive laws:

 (4.3) (4.4)

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,

4.5   Notation (Lines in .) If are distinct points in , I will denote the (infinite) line through and by , and I will denote the line segment joining to by . Hence .

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.

4.6   Example. In the figure you should be able to see the parallelograms defining and . Also you should be able to see geometrically that . What is the point marked x in the figure?

4.7   Exercise. In figure a), , and are the vertices of a regular hexagon centered at . Sketch the points , , and as accurately as you can.

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.)

4.8   Example (Line segment) We will now give an analytical description for a non-vertical line segment , . Suppose first that . The equation for the line through and is

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.

Next: 4.2 Reflections, Rotations and Up: 4. Analytic Geometry Previous: 4. Analytic Geometry   Index
Ray Mayer 2007-09-07