Dot and Cross Products
Suppose [x1,y1] = [ r1 cos(q1), r1 sin(q1)] and [x2,y2] = [ r2 cos(q2), r2 sin(q2)] are points in the plane. Then their dot product
[x1,y1].[x2,y2] = x1x2+y1y2 (dot product definition)
and their cross product
[x1,y1].[x2,y2] = x1y2 - y1x2 (cross product definition)
may be expressed in terms of trigonometric functions and the angles between the two points, or more precisely their position vectors. See below.
Details
To obtain the geometric interpretation, observe the polar and rectangular ways to multiply the first point by the complex complex conjugate of the second, when both are viewed as complex numbers: That is,
[x1,y1][x2, -y2] = (r1 , q1)(r2, -q2)
From the equality of two different ways to multiply points in the plane, observe
[x1x2+y1y2 , x1y2 - y1x2] = (r1r2,q1- q2)
but
(r1r2,q1- q2) = [ r1r2 cos( q1- q2), r1r2 sin( q1- q2)]
Therefore comparison (equality) of real and imaginary parts yields:
x1x2+y1y2 = r1r2 cos(q)
and
x1y2 - y1x2 = r1r2 sin( q)
where
q = q1- q2
is the angle between the two points [x1,y1] = (r1 , q1) and [x2,y2] = ( r2 , q2)
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Complex Numbers |
The fundamental theorem of algebra and partial fraction decomposition in calculus depend on complex numbers.
Easy Consequences
Hint: See the (newest) Complex
Number. Starter Lesson.
for a simple geometric introduction, then continue with easy consequence below.
The clearest geometric proof of the distributive law appears in the
folder.
First Earlier (Old) exposition of complex numbers follows in Z1 to B1 below - read for review or revision .
D1 to D6 after provide a review of vectors.
More on Complex Numbers:
Chapters in Volume 3::
Intro assumes the field properties
of real numbers in place of
deriving them geometrically