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YOU are better than YOU think. Show yourself how:
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-/[]\- Logic
Mastery Do not leave here without it - Logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.
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-/[]\- After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6; For online automated help in senior high school maths & calculus, visit quickmath.com For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com With overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck. |
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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
Hint: See the (newest) Complex Number. Starter Lesson. for a simple geometric introduction, then continue with easy consequence below.
The fundamental theorem of algebra and partial fraction decomposition in calculus depend on complex numbers.
Easy Consequences
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