Coordinate Method
In a plane, the intersection of two perpendicular lines, one horizontal and the other vertical, defines a reference point or origin for the plane. The head of each arrow in the plane has coordinates. Ordered pairs of vertical and horizontal coordinates, ordinates and abscissa, can be employed to add arrows together or find the position of the head of their sum, when the tails of the vectors in the sum are both located at the origin. This gives a fourth method for arrow addition given by the addition of coordinates. This method is very similar to the third method for addition with components. Here is another technical observation with little motivation except for consequences that will follow.
The next figure shows each vector as a sum of components. We assume the vector heads are located at [a,b] and [c,d]. Each vector from the origin may be defined (or drawn) by giving the location of its head.
The next step is to arrive at a formula for vector addition in terms of the coordinates [a,b] and [c,d] of the head locations of the vectors or summands in this vector addition.
The foregoing suggests we represent points in the plane and vectors ending at those points by the coordinates of the head of the vector, that is the coordinates of the points. The foregoing also suggests the vector sum of the arrows with heads at [a,b] and [c,d] respectively has its head at [a+c, b+d]. So to represent the vector addition of arrows, we put
[a+c, b+d] = [a,b] + [c,d]
This gives the coordinate way to add points and their position vectors in the plane. The position vector of a point goes from the origin (its tail) to the point (its head).
See applet for examples. Have it display rectangular coordinates for points and vectors.
Recap and Examples
The sum of two points with the rectangular coordinates [a,b] and [c,d] is given by [a+c,b+d]. We therefore write
[a,b]+[c,d]= [a+c,b+d].
The addition rule is simple add the rectangular coordinates to get the rectangular coordinates of the sum. This coordinate addition follows from our previous discussion of methods for arrow addition.
Example [2,5]+ [6,2]=[8,7].
| 7 !=======+------------//-------* [8,7] | % | [2,5] % 5 |=======* | | | | | | | | | [6,2] | 2 |-------|-//----------* | | | % | | | % | ------------------------------------------------- | 2 6 8 | |
Figure 2. Addition of Points [2,5] + [6,2] = [6+2, 5+2]
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Complex Numbers |
The fundamental theorem of algebra and partial fraction decomposition in calculus depend on complex numbers.
Easy Consequences
Hint: See the (newest) Complex
Number. Starter Lesson.
for a simple geometric introduction, then continue with easy consequence below.
The clearest geometric proof of the distributive law appears in the
folder.
First Earlier (Old) exposition of complex numbers follows in Z1 to B1 below - read for review or revision .
D1 to D6 after provide a review of vectors.
More on Complex Numbers:
Chapters in Volume 3::
Intro assumes the field properties
of real numbers in place of
deriving them geometrically