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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.

parallelgramlaw.gif (7146 bytes)

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. 

Complex Numbers
with easy consequences of two ways to multiply complex numbers in and between vectors & trig, etc

Back ] Area Intro ] Next ]

The fundamental theorem of algebra and partial fraction decomposition in calculus depend on complex numbers.  

Easy Consequences
Vec & Cmplx  No Applet
B2 C. Conjugates
B3 Pythagoras
B4 Distance
B5 Rt Triangle Similarity
B6 Trig., Functions
B7 Dot & Cross Products
B8 Cosine Law
B9 Exponential & cis fns
B10 Easy Trig Identities
B11 Set Viewpoint
Links: Interactive Maths
 

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 Euclidean-Geometry/Complex No.s
folder.

First Earlier (Old) exposition of complex numbers follows in Z1 to B1 below -  read for review or revision .

First (Old) Complex No Intro
Distributive Law
A1 Add Poiints
A2 Polar Coords
A3 Polar Multiply
A4 Complex No.s
A5 Real Numbers
A6 Law of Signs
A7 Key Properties
B1 2nd Mult Method
C1 Unsigned Coords
C2 Signed Coords
C3 Set Codification
C4 More On Real No.s
D1 Arrow Navigation
D2 Sum of Motions
D3 Addition Method I
D4 Addition Method II
D5 Addition Method III
D6 Coordinate Addition
D7 1st Distributive Law
D8 2nd Distributive Law
D9 3rd Distributive Law

D1 to  D6 after provide a review of vectors.

More on Complex Numbers:

Chapters in Volume 3::
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc

This further  Complex Number
  Intro
assumes the field properties
of real numbers in place of
deriving them geometrically


 


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