19-August-2008
Distributive Laws
Plan. Let A, P and Q be points in the plane. The proof of the
distributive law A(P+Q) = AP+AQ will be based
on the observation (the physical assumption) that multiplication by
| A = [r,q]
= [r,0]·[1,q] = [1,q]·[r,0] |
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can be done into two steps. One step is a rotation through the angle q
while the other is a multiplication by the stretch factor or shrinkage factor r
= [r,0]. Multiplication by a stretch factor and rotation through an angle
were shown above to be distributive operations over addition.
Observe that
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| [r,0]·([1,q]·P)+[r,0]·([1,q]·Q |
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| ([r,0]·[1,q])·P
+([r,0]·[1,q])·Q |
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The formula (r1,q1)·(r2,q2)
= (r1r2,q1+q2) implies
(r1,q1)·(r2,q2)
= (r2r1,q2+q1)
= (r2,q2) ·(r1,q1)
due the commutative properties of multiplication and addition with real numbers
(or positive numbers). . Therefore multiplication of points in the plane is
commutative. Thus the commutative law applied to the left distributive
law
A(P+Q) = AP+AQ
term by term, yields the equivalent right distributive law
(P+Q)A = PA +Q A
Products in terms of Rectangular Coordinates
The Key Rectangular Coordinate, Product Calculation Formulas:
- [a,0]·[d,0] = [ad,0] - the real-real case product formula
- [a,0]·[0,d] = [0, ad] - the real-imaginary product formula
- [d,0]·[0,a] = [0, da] - the imaginary-real product formula
- [0, a]·[0,d] = [-ad, 0] - the imaginary-imaginary product
formula
shown with the lesson on Multiplication of Points in the Plane.
The product
| [a,b]·[c,d] = |
( [a,0]+[0,b])·([c,0]+[0,d]) |
by point addition formulas |
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= [a,0] ([c,0]+[0,d])
+[0,b] ([c,0]+[0,d]) |
by the left distributive law
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= ( [a,0] [c,0]+ [a,0] [0,d] )
+ ( [0,b] [c,0]+
[0,b] [0,d] ) |
by the right distributive |
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= ([ac,0] + [0, ad]) + ([0,bc] + [-bd, 0])
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by the key formulas |
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= [ac,ad] + [-bd,bc]
= [ac -bd, ad + bc] |
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The conclusion is that
[a,b]·[c,d] = [ac -bd, ad + bc]
In complex number notation, the latter says
[a+ bi]·[c+di] = (ac -bd) + (ad + bc)i
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Euclidean Geometry
with a geometry based
based development of
complex numbers
24 Lessons:
Correspondence
Isometry
Side-Side-Side
Side Angle Side
Angle-Side-Angle
Isoceles
Right Bisector Construction, Etc.
Perpendicular - Point to Line
SSS Failure
SAS Failure
ASA Failure
Parallel Lines
Angle Sum
Similarity
Right Triangle Similarity
Trig or Similarity
Parallelograms
Kites From Triangles Duplication
Parallelogram from Triangle Duplication
Addition of points in the plane
Multiplication of Points in the Plane
Distributive Law, Step I
Distributive Law, Step II
Distributive Law, Step III
Easy Consequences of this (newest) Complex
Number. Starter Lesson in this site folder follow below.
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
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