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 Read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention. |
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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.
Rotation of a point
Rotation via an angle B of a point with polar coordinates (r, A) adds the angle B to the angle of the point to give an image with angle (r, A+B).
Rotation Distributes Over Addition
Two proofs follow with shortest first.
First Proof - without Coordinates
Examine the following figure with its text, and the material below.
Second Proof - With Coordinates
Assume vectors OP1 and OP2 added together give vector OP3 Assume the heads P1 , P2 and P3 of these vectors have the polar coordinates shown in the diagram.
The origin O and the heads P1 , P2 and P3 form the vertices of a parallelogram. Using the vectors OP1 and OP2 to draw a parallelogram provides the geometric method to add these vectors, a method which results in the vector OP3 as adding rectangular coordinates. Here dashed side P1P3 has length r2.
Figure 2 shows the original vectors OP1 and OP2 and OP3 plus their images under a rotation through an angle B. If we show that origin O and the image vector heads Q1 , Q2 and Q3 form the vertices of another parallelogram then we can conclude that the image vectors OQ1 and OQ2 added together give image vector OQ3 . That in turn implies rotation distribute over vector addition in the plane. Details follow.
First step, we will show that the triangle OP1P3 and triangle OQ1Q3 are isometric (congruent) by the side angle side criteria seen earlier. Here the sides given by vectors OP1 and OQ1 both have length r1. Likewise, the sides given by vectors OP2 and OQ2 both have length r2. Now the included angle for the original triangle has measure A3 - A1 and the included angle for the image triangle has measure (A3+B) - (A1+B) = A3-A1. So the included angles are isometric and the triangles are isometric. Hence dashed side Q1Q3 has the same length as dashed side P1P3 or length r2.
Second step, by similar reasoning, the triangle OP2P3 and triangle OQ2Q3 are isometric (congruent) by the side angle side criteria seen earlier. Hence dashed side Q2Q3 has the same length as dashed side P2P3 or length r1.
Third Step, by the first and second steps, the quadrilateral with vertices given by the origin O and the image vector heads Q1 , Q2 and Q3 has opposite sides equal. Whence the quadraliteral is a parallelogram, and hence vectors OQ1 and OQ2 added together give image vector OQ3 .
In function (like) notation, let RB (r,A) = (r,A+B) define or represent the rotation RB obtained by adding an angle B to the angle part of the polar coordinate (r, A) of a point or vector in the plane.
Then
OQ1 + OQ2 = OQ3 ,
OP1 + OP2 = OP3 ,
OQ1 = RB OP1
OQ2 = RB OP2
and OQ3 = RB OP3
Therefore OQ1 + OQ2 = OQ3 says
RB OP1 + RB OP2 = RB OP3 = RB ( OP1 + OP2 )
Therefore
RB ( OP1 + OP2 ) = RB OP1 + RB OP2
So rotation distributes over addition. Thus the rotating the sum of two vectors (or points) in the plane gives the same result as rotating each vector and then adding. The distributive properties of rotation and scalar multiplication over addition of vectors (or points) in the plane provides the key to complex numbers and using the properties of complex numbers to simplify the exposition of trignometry.
ADVERSTISEMENT
www.whyslopes.com
Analytic Geometry
Area Entrance
Entrance + Pages Below this page
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Area Entrance
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Extras: Not all perfect.
Equal Sign Use/Abuse Real Numbers Say More Positive Linear Inequalities Triangle Inequality Absolute Value |x| |x| Eq'ns & Inequalities Rectangular Coords Shortest Path Distance Formulas Add & Multiply Points Polar Coordinates Radians (A) Vectors (A) Coordinate Arithmetic (A) Navigation on Maps (A) Addition Geometrically (A) Rotation (PT) Translations (PT) Dilatations PT: Rotations
Links to Site Pages outside this site area follow - co- and pre- requisites.
Easy Consequences of this (newest) Complex Number. Starter Lesson follow below to provide an alternate development of HS or college maths.
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 Arithmetic Videos - Real Player Format
Decimal Addition Methods
Decimal Subtraction Methods
Decimal Multiplication Methods
Decimal Division Methods
Fractions
Primes
Greatest Common Divisor
Divisors
Least Common Multiples
Square Root
SimplificationUsing formulas forwards & Backwards - A unifying theme for algebra skill development - the 4th skill in Volume 2, Three Skills for Algebra!