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Distributive Law, Step III
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Euclidean Geometry
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Distributive Law, Step III

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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
A(P+Q)
=
([r,0]·[1,q])·(P+Q)
=
[r,0]·([1,q]·(P+Q))
=
[r,0]·([1,qP+[1,qQ)
=
[r,0]·([1,qP)+[r,0]·([1,qQ
=
([r,0]·[1,q])·P +([r,0]·[1,q])·Q
=
A·P +A·Q

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
= [a,0] ([c,0]+[0,d])
             +[0,b] ([c,0]+[0,d])

by the left distributive law

= ( [a,0] [c,0]+ [a,0] [0,d] ) 
          + ( [0,b] [c,0]+ [0,b] [0,d] )
by the right distributive


= ([ac,0] + [0, ad]) + ([0,bc] + [-bd, 0])

 
by the key formulas
= [ac,ad] + [-bd,bc]
= [ac -bd, ad + bc]

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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