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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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