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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]
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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