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HIP,
HIP, HIP, Hooray
YOU are better than YOU think. Show yourself how:
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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
liinear2007 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.
| |
Aims to adopt to aid, speed & enrich education, yours or
others
The will to sit down and study is needed to enjoy or suffer
through instruction in school and college.
You will miss or postpone opportunities in
later life if you do not learn how to read, write, figure and reason carefully
and precisely. Education open doors. To have more options in education
in skilled trades and professio
plumber, electrician, heavy equipment
operator, nurse, teacher, medical doctor, pharmicists, business operator,
accountant, engineer, scientist
you need to master arithmetic, algebra,
geometry, trig and even calculus or beyond.
Clear aims, say a checklist of what to master sooner or later
for to enter a trade or profession, will help you focus your studies and to work
more efficiently. For an easier time later, you have to find the will to
learn and you need to find out what to study. That being said, you may
meet may skills and concepts in your studies. Several aims or suggestions
are offered below to guide and even speed your efforts in learning.
Education with the will to learn and aims to guide that will to
learn, will we hope go further that studies or education without.
With a list of the skills and concepts you need master, you can sit down and
study them one at a time and after another. The aims stated below point to
skills and concepts you need to master alone or with help for studies in general
and studies in mathematics. The first aim or suggestion below applies to
all subjects. Good luck.
1. Aim for Care and Precision
in reading, writing, reason and arithmetic.
Understanding that carefulness and precision are needed through
will ease or avoid confusion and difficulty in work and studies.
Online logic chapters
2 to 5 from site book Three
Skills for Algebra will test or develop care and precision in reading and
writing.
Testing or developing precision reading and
writing provides a first step in building skills and confidence for work and
studies in many arts and disciplines. It should also the first step for
the greatest benefit from site appetizers and lessons.
While logic chapter 2
to 5 in the online book Three
Skills for Algebra may lead to precision in reading, writing, reasoning,
these online arithmetic review
exercises could lead to precision in arithmetic. Many students are
not aware of the need for results in arithmetic to repeatable and reproducible,
and hence verifiable - that, is they can be checked.
The arithmetic
review exercises need not be done now. When you try them, exercises
involving operations or calculator buttons you have yet to meet can be skipped
until later. See too site arithmetic
videos (realplayer format) to learn or perfect arithmetic skills.
The ability to figure well, without errors, that is in a
repeatable, reproducible and hence verifiable manner, was once considered to be
a sign of intelligence, a sign of skill and competence. It is because a
person who does arithmetic well has learnt to follow multi-step methods with
decimals, fractions or both, and has learnt that lack of care in one step of a
method usually leads to wrong or inaccurate results. So person is aware of the
need for care and precision while following instructions or writing and giving
them. Whence mastery of logic and learning to figure well all imply or
demonstrate greater care and precision for work and studies, and so point to
fewer difficulties for both work and studies. Take note.
Students: When you were young, to read,
write and spell, you learnt all the alphabet not just some. With
regrets, algebra and higher mathematics demands mastery of times tables,
arithmetic operations with decimals (addition, subtraction, multiplication,
long division) and with fractions. During arithmetic mastery, you learnt or
will learn that an error in one step of a multi-step calculation usually leads
to wrong results. So each step, the work, has to be done with care and
precision. The same or similar care is needed in algebra, logic and beyond.
Parents: If your teen has learning
difficulties in general or in mathematics, see if your teen can master logic
alone or with help. Also emphasize for your teen, drill and practice in
arithmetic, until the result are almost always correct. Mastery of logic
and mastery of exact arithmetic with whole numbers, if that be possible,
will show your teen the need for care and precision in 4Rs: reading, writing,
arithmetic and reason. That should reduce learning difficulties in general and
in mathematics. Good luck. Encouragement of figuring skills, until they give
repeatable and reproducible results can begin in primary school.
Teachers: In mathematics and many other
subjects, students first need to learn how to use rules and patterns, one at a
time, and one after another, with precision, so that you obtain same results
as others. If the results obtained are not repeatable and reproducible,
students are not being careful enough. Have them slow down. Rules and patterns
need to be read and applied with care and precision. After that students
may learn to combine rules and patterns to obtain or produce further rules and
patterns, again in a repeatable and reproducible manner.// Students skills and
knowledge of mathematics may mix (a) rules and patterns that they
have been given or found) with rules and patterns obtained from
combining the given or found ones. When or if students become
comfortable with of obtaining or deriving new rules and patterns from
given ones, they you can enjoy a thought-based understanding of
mathematics to come and, optionally, mathematics seen
earlier. The latter option in part or full may be too
detailed for the liking of most students, students who will be content with a
thought-based understanding of the mathematics to come. However, the
option to obtain a thought-based understanding of mathematics from arithmetic
to calculus and beyond, subject to the limitations of pattern
based reason, should be available to students in texts online and
off for students keen to know or keen perfect their skills and
knowledge. What is important is that students learn to apply and
combine rules and patterns in a repeatable and reproducible manner from
primary school to college level. // To Learn More, see item 7
below. Three
Kinds of Reason.
Training future mathematics teachers in the
UK and North America that correction of errors is harmful to the ego of a
student undermines the duty and need to teach students to figure
efficiently. Teaching training colleges would serve future students better
by emphasizing arithmetic and algebra skills, and how to develop them
instead of technology.
2. Aim for Fraction and Algebra Skills and Sense
The careful and precise use of whole numbers and fractions in
solving linear equations will provide a firmer base for further mathematics
instruction. Some steps can be followed or designed to avoid integers and
signed numbers. Emphasizing that solutions of linear equations can be
checked before being submitted for correction or marking will eventually lead to
greater student independence. When a check fails, the mistake or mistakes
fall between the start of the solution and the end of the check.
For all high school and older students, the unique site area solving
linear equations introduces and illustrate fractional ops on line segments
(sticks) to geometrically develop algebra and fraction sense and skills. That
with solving triangular and essentially-one-unknown simultaneous equations
begins a new, yet tried and tested path to make solving word
problems, substitution, the distributive law, and solving simultaneous
equations much easier learn and teach. The geometric approach is intended
to promote the algebra approach, and it usually does. Where it does not,
students will need further help. Fractional operations on stick diagrams to
build algebra and fraction skills and sense is a site invention - a
co-invention with an offline author.
While some students will want to leap into the algebraic
way of solving linear equations, staying a little longer with or leaving and
returning to the stick diagram approach will reinforce fractions skills and
sense. The fraction parts of the site area Fractions,
Ratios, Rates, Proportions & Units offers
fraction skills by rote,
automation is important, and also fraction skill development with more
operations on line segments to provide and perfect a thought-based
mastery. The latter is recommended only for (i) students who want to
learn, and enrich or perfect their skills and knowledge, and (ii)
students who dislike or want to go beyond rote learning of methods. That
being said, drill, practice and correction with fractions and solving linear
equations may build skills and confidence through the mastery of methods with
repeatable, reproducible and hence verifiable results.
3. Aim for a Greater Use of Words in Mathematics
Many people have difficulty with algebra. Even
gifted or advanced students will have gaps in their understanding of the
shorthand role of letters and symbols in mathematics. After all, arithmetic
and algebraic expressions, the longer ones, are difficult or impossible to
read aloud, symbol by symbol, bracket by bracket, and so on. And when
expressions are read aloud, there is the frustrating possibility of confusion
in the order of operations. The reader may mean one thing, and the
hearer may write something else. That is not good for precision in reading and
writing in arithmetic and algebra. This difficulty in reading arithmetic and
algebraic expressions aloud, including those appear in the algebraic described
properties of arithmetic with numbers, leads to silence, a dearth of
words, in skill development and communication. We look at and digest
mathematical expression in a glance instead of reading them aloud. Mathematics
in and beyond arithmetic and algebra has been a silent discipline where
written letters and symbols, together or separately, are used for
communication. That makes learning and teaching harder.
Site book Three
Skills for Algebra with its online postscripts point to a greater and
clearer use of words in mathematics in two different, but mutually supporting
ways,
-
Greater Use of Words in Describing Numbers, Amounts and
Quantities: Algebra
chapters 8 and 9 identifies our ability to describe numbers, amounts and
quantities with written or spoken words before or besides symbols. The
online postscripts use words to explain what
is a variable, constant, or parameter, and do so without the use of
symbols in a manner that students can grasp years before functions, a
senior high school topic, are met. The online postscript brings
mathematics in closer alignment with physics where numbers or quantities
that may vary in one sense or another, are called variables.
-
Greater Use of Words in Common Operations on Equations
and Formulas: Chapter 14
on compound interest or growth formula introduces the direct and
indirect use of formulas or equations, and shows the difference between algebraic
and numerical solutions, and point to the ability of the algebraic
solution to give the numerical solution in full, or just the end
results.
For all formulas in high school and college mathematics, we
may now identify (A) direct and indirect use, and the (B) numerical
and algebraic solutions that may be possible in the indirect use.
The repeated use of two phrases in dealing with formulas, one at a
time and one after another, gives voice to a previously unnamed and hence
hidden operations and themes in mathematics learning and teaching. Remembering
the phrases (A) and (B) while you study or teach will make the methods
of algebraic ways of reasoning clearer and provide a focus or two for their
study.
Words and names are powerful. Once an
mathematical object or operation is named, and clearly described with
words, we can use names and words, again and again to point out
recurring patterns. The foregoing lessen the silence that accompanies
arithmetic and algebraic expressions, formulas included, because they are so
awkward to read aloud term by term, parentheses by parentheses. The foregoing
introduces a new avenue for mathematical learning and teaching. Formula,
operations and properties of real numbers, etc, known and named can be
mentioned and discussed without being present in written form. The
result is or will be more written or spoken communication in mathematics based
on words to supplement or go beyond expressions better seen and read silently
at glance, than read aloud in a way that communicates order of operations
clearly and precisely.
4. Develop More Algebraic Thinking Skills, Those Needed
for Calculus
Preparation for the full strength use of algebraic ways of
writing and thinking in Calculus. The algebraic way of writing and reasoning is
required at full strength in calculus. This full strength requirement, and
algebra shock, is too sudden and terminal for many students. This aim (and
the previous ones) point to a remedy, one that many, not all students have
enjoyed.
Calculus gives the best framework for
understanding calculations met in business, science and engineering.
Describing the same calculation without calculus is long and shallower
process. Shortcomings in the development o algebraic skills and concepts,
those needed for calculus, led schools and course design to fill student with
topics not needed for calculus and supposedly simpler. Good preparation for
college mathematics (or calculus) requires mastery of most, but not all
the topics, you meet in high school mathematics: exact arithmetic with whole
numbers and fractions, algebra, geometry without and with coordinates, and
trig.
Calculus in the first instance is a subject of slope and rate
related calculations, as is or reversed, with applications.
The online version of site Volume 3, Why
Slopes and More Mathematics, includes a geometric
calculus preview before a more algebraic perspective in chapters
2 to 6 . The geometric
calculus preview explain how slope related calculations, forward, not
reversed, appear in calculus. That gives context or explains why slopes
appear repeatedly in earlier high school and college mathematics. In an
courses where slopes and then polynomials and rational functions are
met, the geometric calculus
preview and chapters 2 to 6
could be used to (i) understand and explain extreme points and
identify where factored or easily factored polynomials and rational functions
are increasing or decreasing; and (ii) to develop students algebraic reasoning
concepts and skills. The foregoing also provides a way to ease or
avoid difficulties in the first and further weeks of calculus.
Calculus is also the subject in which the many facets of what is
a variable, constant and parameters appear as well. That being said, chapters
2 to 6 in Why Slopes and More
Mathematics provide a slow and effective, induction, into the algebraic way
of writing and reasoning required.
Chapters
2 to 6 with the aid of slope interpretation identify interior and
end-point extreme points (maximums and minimums). Polynomial and rational
function formulas given for slopes (they are not computed in these chapters),
and given in factor formed, are used in slope sign analysis. The sign
analysis of these factored polynomials and rational functions indicate the
intervals where a function y = fix) is increasing or decreasing, and thus
indicates the interior or end-point location of extreme points. By
skipping over lengthy discussion of limits and derivative calculations to the
sign analysis of derivatives or slopes, these chapters provide a context for
the skipped material while developing the algebraic maturirty needed to
understand the skipped material.
The following methods or path in chapters
2 to 6 for easing or avoiding algebra shock in calculus may be seen at the
start of calculus and prior to that, in course in analytic geometry after or as
part of the discussion of slopes to straight lines and the factorization of
polynomials alone or in the numerators and denominators of quotients (rational
functions).
5. Aim for a Greater Use of Complex Numbers
(this could come before calculus)
This aim points to a change in course design and delivery at the
secondary and tertiary (college level). Senior high school and college
students may use these underlying ideas in their self-instruction. Course design
changes indicated here will most likely not occur in their school days.
A simple and clear way to understand and explain complex
numbers (site starter lesson, pre-calculus level) is to introduce addition
of points in the plane using rectangular coordinates; to introduce their
multiplication via polar coordinates; and then to assume or geometrically
imply the arithmetic properties of complex numbers. Implicit here is the
assumption, that every point in the plane has both rectangular and polar
coordinates. From the numerical properties of complex numbers,
algebraically described, we can obtain several easy consequences: a new proof
or confirmation of the Pythagorean theorem, the properties of trigonometric
functions; and a geometric, complex number development of trigonometry.
Details are given in the site area on complex
numbers. All the foregoing suggests simpler path for high school
trigonometry and simpler, complex number developments of trig expressions for
dot and cross-products of vectors in the coordinate plane. University
level schools of engineering and science will appreciate the shortcuts. They
can be also be used in senior high school mathematics before calculus if time
permits besides the other curriculum obligations.
Remark 1: Teachers will find several approaches to the
derivation of the arithmetic field properties of complex numbers in site
pages. All assume points in the planes drawn on paper can be represented
by ordered pairs of rectangular and polar coordinates. All stem from
different assumptions about Euclidean geometry, the use of coordinates and/or
real numbers. The most extreme route is to (i) assume decimals can be
used as coordinates, (ii) assume the completeness of this representation by
accepting infinite decimals expansions as coordinates, and (iii) imply or
derive the properties of real numbers from assumptions about geometry and the
use of unit lengths and directions to define coordinates systems. The
latter route employs geometry instead of set theory to represent and obtain
the properties of real numbers. The latter route met first could provide
a context for modern mathematics even though modern is context-free in another
sense.
Remark 2 : Pure mathematics from the algebraic
statement of axioms (assumed patterns) provides a thought-based
development or codification of concepts and statement in mathematics with no
logical dependence on suggestive drawings nor physical argument. So pure
mathematics is said to be context-free , but there is a context for this
context-free development. In contrast to pure mathematics, applied or
mixed mathematics employ suggestive drawings and physical arguments to
introduce and employ coordinates or other device from pure mathematics to
model objects in space and time. That steps beyond pure mathematics,
while assuming those calculations in the model which can be done in a
mathematics or context free (formally or purely) way in order not to
accidental introduce further physical assumption into the calculations or the
results. Finally, primary, secondary and tertiary mathematics from
learning to count and recognizing geometric shapes to the definition of trig
functions using right triangles and unit circles in a coordinate plane is not
part of pure mathematics. Before pure mathematics can begin, mathematics
education has to inductively (use of manipulatives and suggestive drawings
appears here) develop the necessary numerical, algebraic and
deductive skills and sense. The set theory axioms of pure mathematics
imply the axioms about real numbers which appeared in modern mathematics
course design. But there was a mistake. Modern mathematics course design
did not mention decimals and the emphasis on context-free development was
inconsistent with the high school and college development of Euclidean
Geometry, Trig and Calculus with the aid of suggestive drawings.
Modern mathematics course design inserted set language into
course design - that description awkward in parts can be still be retained
and refined as it provides a finer or more precise language for the
development of some skills and concepts. How will be explained
While the logical development of pure mathematics is and
should be context-free to avoid dependence on suggestive drawings or physical
assumptions, both of which can be useful but misleading, pure mathematics
itself codifies the properties of real numbers used as a coordinates in mixed
or applied mathematics. Comprehension of how and why provides a context
for the avoidance of suggestive drawings and physical arguments in pure
mathematics.
For success in mathematics, the instruction of
students needs to emphasize an operational command of arithmetic and develop
the ability to use rules and patterns one at a time and in combination, one
after another, not only for an operational command of mathematics, but
also for its comprehension. Students who learn how to combine rules and
patterns, rules and patterns mastered by rote if need-be in the first
instance, in seeing how to combine may grasp the logical
development of further mathematics and may also, if interested, see the
logical development of the rules and patterns assumed or mastered by
rote - the unavoidable starting point for mathematical skill and concept
development.
There are three kinds of rule-based intelligence in mathematics,
logic and most pattern-based subjects.
-
The first kind met in primary & secondary school
arithmetic consists of skills with repeatable, reproducible and therefore
formally verifiable results - results that are then considered right or
wrong. This kind of reason can support the mastery of arithmetic with
fractions, whole numbers, real numbers and complex numbers; and beyond that
the mastery of solving linear equations, the use of trigonometry and
geometric reasoning. The first kind of reason can also be useful in money
matter (the buying and selling of services) and in many trades and
professions.
-
The second kind also met in primary school consists of
pattern or rule recognition. The development or exploitation of the ability
to recognize or suggest simply patterns in order to predict the next element
in a sequence. If the prediction fails, another pattern is required.
-
The third kind, assumption-based, deductive reason, appears
after inductive mastery of logic, that is mastery of implication rules If A
then B and their use. The third kind combines and applies implication rules
and definitions and assumptions, one at a time and one after another, to
arrive at logical conclusions. Here chains of reason how to be posed in a
readable, repeatable, reproducible and therefore a formally verifiable
manner. After mastering the use of rules and patterns first alone and
then in combination, the latter combination of assumed rules and patterns to
produce results or further rules and patterns provides the possibility of a
thought-based development and structure for skills and concepts to be meet,
if not previously met. Explanations for methods previously mastered in a
repeatable, reproducible and thus verifiable may here be available as an
option, but it should not be forced on students to avoid alienation.
For third kind of thinking in mathematics, there was a search
for secure assumptions, so that deductive reason could proceed in a
consistent and reliable manner. Unfortunately, uncertainty results
in mathematical logic imply more can suggested than proven in mathematical
theories which are not finite. So the assumptions made for the third kind of
reason stem from experience or trial and error over time. That identifies modern
pure mathematics as another empirical art. But mathematics by providing a
format for measurement and calculations remains the queen of
science, a queen in the hierarchy of empirical arts.
Pre-coordinate Euclidean geometry, the original model
for pure reason in mathematics, with its assumptions and deductive chains of
reason is still worth presenting in part if not in full in high school
mathematics in a selective manner to build algebraic-deductive skills and
geometric skills and sense. However, the empirical nature of pre-coordinate
and hence coordinate-free Euclidean Geometry is implied by diagrams with
subtle faults that imply incorrect conclusions - subtleties detected with the
use of coordinates in advance mathematics courses.
7. Try to Understand the Decimal Viewpoint of Limits,
Convergence and Continuity.
Bring Back, O Bring Back, the Bonny Decimals.
Calculus and mathematics instruction became more complicated
in the second half of the 20th Century due to the course design which employed
decimals to represent numbers and to do calculations, but which also did not
favor nor mention them in presenting theory. In particular axioms for
real numbers (algebraically described properties of arithmetic with real
numbers represented by decimals) do not mention decimals in high school
mathematics despite coverage of scientific notation and decimal-baed accuracy
or estimates in arithmetic. Then in calculus, hard to grasp, extremely
algebraic, decimal free explanations of limits, continuity and convergence are
inaccessible. As a mathematics students, the site author Professor
Whyslopes, struggled repeatedly to understand the decimal approach in
technical and intuitive manner. For many students, that decimal free
approach is a very large barrier to understanding. So today, calculus courses
may teach students the properties of limits, continuity and convergence by
example (rote learning) and not attempt to provide any theoretical framework.
The modern mathematics description of limits,
continuity and convergence is often decimal free. The modern mathematics
curricula of the 1950's and 1960s, echoes of which still appear in some or all
classrooms today, did not mention nor sanction the decimal representation of
real numbers and so did not sanction methods of decimal arithmetic, while
still employing in numerical results. Moreover the decimal perspective of
limits, continuity and convergence was not mentioned or sanctioned as well.
Limits, Continuity and Convergence in Calculus.
A decimal viewpoint of limits, continuity, and convergence,
and the associated question of limited or unlimited error control in function
evaluation or computations, is sufficient for most students and its
provide an model which also makes the decimal -free viewpoints easier to
understand and grasp - provides a context for the latter. Therefore chapter
14 in Why Slopes and More
Mathematics introduce the decimal viewpoint while the appendices
to this volume push (or review) the decimal into advanced calculus or real
analysis. That provides the proofs of theorems often given without in first
and further courses in calculus.
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whyslopes.com
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HIP, HIP, HIP, Hooray for
site
content & history. Hype, Hype,
Hype, Hoorary, for deception.
Your IP Address & how to use
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(i) First
Year High School Math - Lesson Plans with Fraction Focus
(ii) Second
Year High School Math - Lesson Plans with an algebra focus
(iii) Algebra
Lesson Plans
Help U Learn/ Teach
- Algebra
words before symbols
- direct & indirect
use of formula, numerical versus algebraic solutions - what
is a variable (more words)
- Arithmetic
- exercises
- with fractions
-
videos on primes, lcm, gcm,lcd, square roots etc
- Calculus - geometric
preview, algebraic
preview,
3 study guides,
much more
- Complex numbers
-starter lesson with java applet - easy
consequences for trig & vectors in the plane
- Education
- Empirical Course Design
& Delivery
- Fractions
- alone
- by rote
- with
algebra
- videos
- Functions - introduction
hindsight
- composition aka
substitution -
- Geometry, Euclidean - Correspondence
of triangles, Triangle
construciton, duplication & Isometry - Failure
of ASA & the // line postulate - angle
sum in triangles -//
grams - Triangle
Similarity
- Geometry-
Analytic - functions, polynomials, complex numbers, unit circle
trigonometry
- Logic
- First Steps -
Symbols in Logic
-
Occurrence &
Truth Tables - Indirect
Reason -Indirect
Reason More
- Proportionality
- Definition -
Direct & Indirect Use - Numerical versus Algebraic Solutions
- Real Analysis
- Decimal View of concepts
and of proofs
- Rules &Patterns in Science, Technology & Society
- Pattern Based Reason
- Mathematical Reasoning, empirical, inductive or deductive
- Units
- in rates & slopes &
(?) derivatives
- in ratios
& proportions - slopes & rates included
- Complex Numbers & Vectors & Trig
- trig expression for dot
& cross - cosine law
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