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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.

wpe1B.gif (3319 bytes)

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.

wpe1D.gif (7585 bytes)

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]

 

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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