YOU are better than YOU think. Show yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work |
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Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, 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 liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
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.
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)
www.whyslopes.com
Complex Numbers
HInt: See the (newest) Complex Number. Starter Lesson. Then continue with easy consequence below.Easy Consequences
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 Links: Interactive Maths First Earlier (Old) exposition of complex numbers follows in Z1 to B1 below - read for review or revision .
First (Old) Complex No Intro Distributive Law A1 Add Poiints A2 Polar Coords A3 Polar Multiply A4 Complex No.s A5 Real Numbers A6 Law of Signs A7 Key Properties B1 2nd Mult Method C1 Unsigned Coords C2 Signed Coords C3 Set Codification C4 More On Real No.s D1 Arrow Navigation D2 Sum of Motions D3 Addition Method I D4 Addition Method II D5 Addition Method III D6 Coordinate Addition D7 1st Distributive Law D8 2nd Distributive Law D9 3rd Distributive Law D1 to D6 after provide a review of vectors.
More on Complex Numbers:
Chapters in Volume 3::
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc
This further Complex Number
Intro assumes the field properties
of real numbers in place of
deriving them geometrically