Parallelogram Method
When two arrows or vectors (representing motions if you wish) have a tail at the same place, they may be added together by moving the tail of one to the head of the other with the aid of a parallelogram, and then using the head to tail method for addition. This gives the parallelogram method for adding a pair of arrows or vector addition. The resultant arrow does not depend on which arrow, the first or second, is moved. Here is a repeatable and reproducible methods, arbitrarily defined, for vector or arrow addition. More generally, parallelograms can be used to displace or move arrows from one location to another without changing their lengths or directions. But that is another story, or chain of reason, not illustrated here.
The following diagram shows in four steps 1, 2, 3 and 4 how to add two vectors which start at the same place, or have the same origin.
Step 5 just gives the shorthand notation for a vector from a point A to B in a figure. Outside of a figure, I will use the notation AB to denote the same vector because it not easy to put an arrow over the two letters AB. The parallelogram law says how to compute AB + AC.
Saying how to add vectors or "straight-line movements" which have tails or starting points defines a second kind of vector addition. The first first kind, head to tail addition of vectors representing movements, was described in the previous lessons. In mathematics, you should remember saying how to do a calculation or operation defines it. Remember this principle. Vector addition represents as given here represents a geometric calculation.
From the parallelogram rule or law for addition, the calculation of the sum of vectors AB + AC and the calculation of AC + AB is identical. So the order of addition is not important. Expressions of the form AB + AC can be replaced by the expression AC + AB when we describe calculations with vectors (arrows). That is, vector addition commutes. Remember this when we talk about real and complex numbers below.
The applet illustrates the parallelogram law
for the addition of vectors with tails at the origin.
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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