AU: tutorfinder.com.au CDN : findatutor.ca CDN: .i-tutor.ca CDN: Montreal Tutors NZ: findatutor.co.nz UK: tutorhunt.com UK: tutors4me.co.uk USA: wiziq.com USA: ziizoo.com |
|||
|
YOU are better than YOU think. Show yourself how:
|
-/[]\- 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.
|
-/[]\- 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. |
|||
|
Explore collaborative whiteboards from groupboard, twiddla or scriblink. |
How to Multiply I
Next we define using polar coordinates the product of two points in the plane using the the add the angles and multiply the lengths rule. The product of a pair of points with polar coordinates (r1,q1) and (r2,q2) is
|
Note: In the case where r1r2 = 0, we will take the product to be points with rectangular coordinates [0,0] and polar coordinate (0,0).
Example. Two arrows are to be multiplied. One has length 1.3 and angle 22.62°; the other factor has length 1.026 and angle 46.97°; and so their product has length 1.3338 = 1.3·1.026 and angle 69.69° = 22.62°+46.97°; and that is it. See the following diagram.
Another Example. The product of the two points (3,80°) and (4, 60°) is ((3)(4), 80°+ 60°) = (12,140°)
Technical Details: Algebraic Properties (or more precisely, an algebraic description of computational properties: The commutative and associative laws for the addition of points in the plane follows from the corresponding laws for addition of real numbers.
Commutative Law for Products of points in the plane
whenever (r1,q1) and (r2,q2) are polar coordinates for a pair of points in the plane. This property follows as the commutative law of addition for real numbers (or angles > 0) implies q1+q2 = q2+q1 and the commutative law for products of real numbers (> 0) implies r1r2 = r2r1,.
(r1,q1)·(r2,q2) = (r2,q2)· (r1,q1) Commutative Law for Products of points in the plane.
whenever (r1,q1), (r2,q2)}and (r2,q2) are polar coordinates for a pair of points in the plane. This property follows as the associative law of addition for real numbers (or angles > 0) implies {q1+q2}+ q3= q1+ {q2+ q3} and the commutative law for products of real numbers (> 0) implies {r1r2}r3 = r1 {r2r3 }
{(r1,q1)·(r2,q2)}· (r3,q3) = (r1,q1)·{(r2,q2)· (r3,q3)} Multiplicative Inverse for non-zero points in the plane.
if (r1,q1) is the polar coordinates of a nonzero point in the plane.
(r1,q1)·( [1/r1] , -q1) = (1,0) Technical Detail within a Technical Detail: There is a geometric viewpoint of points and vectors (or arrows) in the plane which implies the commutative and associative laws or properties of addition for both points and real numbers.
Summary
The addition of points in the plane is given by means of their rectangular coordinates while multiplication is given in terms of polar coordinates. Below you will see how to multiply points together using rectangular coordinates as well. The equality of different ways to multiply points together leads to many properties of vectors and trigonometry.
|
|
www.whyslopes.com
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