19-August-2008
Zero-Degree Multiplications
(scalar multiplication by unsigned numbers)
Recall (r1,q1)·(r2,q2)
= (r1r2,q1+q2)
In the case q1= 0, the polar
coordinate, product formula yields
(r1,0)·(r2,q2)
= (r1r2,q2)
Now
(r2,q2) = [a,
b]
in the sense that the Left-Hand-side and the right hand side determine the
some point in the plane. That assumption stems from and exploits an extrinsic
view of the plane and the use of rectangular and polar coordinates. The
line segment [0,0] to [a,b] has length r2 and angle q2
with the positive direction of the x-axis. It provides the hypotenuse of right
triangles with legs on the axes or parallel to the axes and meeting the point
[a, b]. The point [ r2a, r2b]
determine similar right triangles with a hypotenuse that makes the same angle q2
with the positive direction of the x-axis. Similarity of right triangles
implies the point [ r1a, r1b] is at distance
r1r2 units to the origin. So
(r1r2,q2) =
[ r1a, r1b]
The latter formula provide an alternate means for computing the product (r1,0)·(r2,q2)
using the rectangular coordinate [a,b] of the point (r2,q2)
= [a, b].
First Scalar Multiplication Distributive Law
The formula
(r1r2,q2) =
[ r1a, r1b]
implies multiplication by points (r1,0) = [r,
0) where r > 0 distributes over vector addition
since [ r1a, r1b] + [ r1c,
r1d]
= [ r1a+r1c, r1b
+ r1d]
= [ r1(a+c), r1(b + d)]
= r1[(a+c), (b + d)]
= r1 ([ a, b] + [c, d]
)
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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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