See 9. Urval med återläggning

The basis for all combinatorics.

An array of numbers.

The "sum" of an array of numbers.

You can expand (x + y)ⁿ as follows.

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This is not the same thing as (xⁿ + yⁿ)! That is a sum of e.g. cubes like x³ + 1, where the exponents are "inside the parentheses", for which there is a different formula (Sum/difference of cubes).

    1       zeroth row -- a constant: (x + y)^0
   1 1      first row --- a binomial
  1 2 1     second row -- a binomial to the second degree
 1 3 3 1    third row
1 4 6 4 1   fourth row

Finding combinations of k items from n, node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 "n choose k", is node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

For example, node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

A startling symmetry is that node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 which is the same as the previous identity. This symmetry, that node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 works for any numbers node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 and node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 you want.

A combination lock should be called a permutation lock. The order matters, 2-7-1 isn't 2-1-7.

There are more permutations than combinations.

Note that 10! / 8! is the same as 10 * 9. If it's not obvious why, think.

• Math

The LCM is called a MGN (minsta gemensamma nämnare) in Swedish because it's for use in a denominator. You have several fractions with different denominators and want to expand them to use the same. In principle it doesn't have to be about denominators, so call it LCM.

To find the LCM of two numbers, construct a number that contains all the prime factors of both.

Bigger than the Greatest common factor (GCF), in the sense that you use every prime you have access to (though not in their full multiplicity). Note the naming of LCM versus GCF. "Multiple" refers to a product of possibly many numbers, "factor" refers to something you can factor out of a single number.

To find the GCF of two numbers, find what primes they share.

Use Divisibility tests or the Euclidean algorithm.

Smaller than the Least common multiple (LCM) in a sense, that you don't use all the primes you have access to. In reality it's apples and oranges. The LCM is to expand two fractions to have the same denominator, so that you can then add them. The GCF is to deexpand a single ratio (fraction or equation) to its most simplified form.

In short, you shrink a denominator to the GCF, or you expand a denominator to the LCM.

See Anki cards. Basically |x| + |y| ≥ |xy|, because x and y, being vectors or complex numbers, take a long way around while xy takes the shortest way.

The rules are braindead. This wand just works everywhere.

(A)^B mod C ≡ (A mod C)^B mod C

(A + B) mod C ≡ A mod C + B mod C

(A * B) mod C ≡ A mod C * B mod C

You've used long division. You're familiar with getting a quotient and a remainder? This is just a realization of that, which we can then use for other things. Warning: the polynomial q(x) has an unfortunate name, it is not the quotient. Without further ado, realize that:

for two polynomials p(x) and q(x), there are unique polynomials r and k such that

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which can be written

node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

which is the same as

node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

See the Polynomial Remainder Theorem on how to use this last form.

Divide the polynomial p(x) with (x - a). The remainder will be identical to the value of plugging a into the original: p(a). This lets us know all sorts of things. If the remainder is zero, then (x - a) did divide the polynomial cleanly, and a is a root.

Restate the original Division algorithm, node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

in these terms. New division algorithm: node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

Which leaves only k(x) as the unknown.

Let's find GCF(209,77). Realize that 209 can be restated in units of 77, plus something to make up the remainder.

209 = (2 * 77) + 55

This rewrite is the basic step. Repeat: restate 77 in terms of the new thing.

77 = (1 * 55) + 22
55 = (2 * 22) + 11
22 = (2 * 11) + 0

The algorithm is: Re-apply division until you get zero remainder. Then the divisor you just tried is the GCF.

Diophantine equations come in many forms. Fermat's Last Theorem is a famous example. What we mean with this word, 'Diophantine', is that we're interested in integer solutions only. This is why it has applications for e.g. proving the Fundamental Theorem of Arithmetic, which is about prime numbers, which of course are integers. We're going to solve a simple kind, the linear Diophantine equation.

Note: it may seem pointless to run the Euclidean algorithm when you can detect that two numbers share no common factor. But you do it for the steps in the process, not the end result (that GCF=1).

Take an equation of the form ax + by = c, where a, b, and c are known integers, and we're interested in integer solutions. For example 43x + 19y = 4.

Normally equations of this form describe a continuous line, crossing an infinite number of points in the real plane, and some such equations even manage to skip any integer pair (x, y).

Crux: For there to be an integer solution to ax + by = c, then all GCF(a,b) must be a factor of c. If a and b have a common factor not shared by c, then stop. There are no integer solutions. This means you can't e.g. have even numbers for both a and b and an odd number for c, but one of a or b may be even, and c may be too.

Aside: a statement (ka + lb) is a linear combination of (a + b). Therefore, ax + by is a linear combination, too. You can put in any x and y. Anything that divides (a + b) will retain its ability to divide (ax + by).

All numbers are either primes, or a combination of primes.

Minimally stated: Every positive-degree polynomial has a complex zero.

Alternatively: An nth-degree polynomial has n zeros, if you include complex zeros and count multiplicity (duplicate zeros).

That follows from the minimal statement, because once you know you have a zero, you can factor it out and are left with a new (lower-degree) polynomial, which of course also has a zero.

• Divisible by 2 if it's even.
• Divisible by 3 if the sum of the digits are.
• Divisible by 5 if it ends in zero or five.
  • An odd number containing the factor five will end in five.
  • An even number containing the factor five will end in zero.

domain and range

• take a function f: A → B
  • the domain is A, can refer to it by D_f
  • the målmängd is B
    • the range (värdemängd) V_f is a subset of B

find the range of a function

• see if it has a min or max

The range of a function that only takes on the values 1 and 2 is written {1,2}, not [1,2], which would be the full interval.

A closed interval [a,b] includes the ends. node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 .

[a,b]  --------[a----------b]--------

An open interval ]a,b[ excludes the ends. Unfortunate notation. node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 .

]a,b[  ---------a[--------]b---------

Memory key: the sum will have a binomial factor that contains a plus sign, just like the sum itself.

a³ + b³ = (a + b)( … )

a³ - b³ = (a - b)( … )

Solve x³ − 125i = 0 for node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 .

First, the notion of algebraic numbers: a number that can be a solution to some equation with rational coefficients. This would be most numbers that you actually use.

One example is √2 because it is the solution to x² - 2 = 0. It is algebraic.

Though there are many more transcendental numbers than algebraic, it is hard to prove that a number isn't algebraic. Examples of proven transcendentals include π and e.

• Math

Addition and subtraction work for matrices of identical dimensions.

Multiplication works when the "inner two" dimensions are the same. Meaning: columns of first matrix = rows of second matrix.

The matrix multiplication A_3x2 times B_2x3 is defined. Putting the dimensions next to each other: 3x2 2x3 The "inner two" are the same.

To multiply, a visual aid is to raise the second matrix:

    1  0  0
    3  2  1
2 3
1 0
4 1

Then, for the first item, add the product of 2*1 and 3*3.

An identity matrix I, in Swedish denoted E, looks like this. The zeroes are shorthand for "there are zeroes all over here".

1     0
  1
    1
0     1

A square matrix has two possible properties:

• EITHER its determinant is zero
• OR it is invertible

Take two square matrices A and B. If AB = I and BA = I, they are inverses to each other. The inverse is unique. The inverse of A is denoted A^-1.

Powers function like you expect: (AB)^-1 = B^-1 A^-1 but only square matrices

Calculating a determinant - Sarrus' rule, Laplace utveckling, or Gauss-Jordan elimination

Calculating the inverse of a 2x2 matrix: determinant (ad-bc) times the adjugate.

Calculating the inverse of a 3x3 matrix:

Your original KhanAcademy method involves the determinant, the matrix of minors, the cofactor matrix and the adjugate matrix.

• Matrix of minors is nine 2x2 determinants.
  • This produces something called the cofactor matrix.
• Taking the transpose of the cofactor produces the adjugate matrix.
• You know this. 1/det(A) times adj(A) produces the inverse to A, same as for 2x2 matrices.

The Gaussian method

• Draw your matrix. Draw a vertical line. Draw the identity matrix on the other side.
• Do elementary row operations, and whatever you do to the first matrix, you have to do to the identity matrix.
• Goal: the first matrix should become the identity matrix.
• Result: the original identity matrix has become the inverse matrix.

Take a couple of 2x2 matrices, call them A and B. Say that

det(A) = -1 and
det(B) = -3.

What are the following?

det(AB) det(2A) det(A⁺) det((B²)^-1) det(B⁵)

How to calculate a determinant:

You can Gauss it until you have a triangular matrix. Then the determinant is the product of the diagonal elements.

For a 2x2 matrix, the determinant is node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 .

Write a system of equations as a talschema (this is not a matrix). Reduce.

Elementary row-ops:

1. Swap two rows
2. Multiply row with something
3. Add the expanded row to another row

Reduced row echelon form -- trappstegsform.
1 * * * * | *
    1 * * | *
      1 * | *
        1 | *

Each 1 in the diagram above is a "pivot".

Three possibilities

1. There is a row 0 0 ... 0 | a. No solutions.
2. There is a pivot in every column. One unique solution!
3. There are columns without pivots. Infinite solutions.
4. If there is a row 0 0 ... 0 | 0, you can just toss that row. 0=0 is true but adds no info.
5. If there are two identical rows, you can toss one of them. It adds no info.

Warning: Can have node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 as vectors, but node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 as directed distances (riktade strackor).

Subtraction is tricky: get the order right. Take three points A, B, C. AB minus AC is what? Look at the minuend(?), AB. The thing being subtracted from is AB. Therefore, AB is pointing at the same point that the resulting vector will point at. So the resulting vector is CB: it goes from C to B.

If you ever have coordinates X, Y, neither of which are in the origin, you can point at either of them with vector addition. Take a point O in the origin and add OY to YX to get OX. Or add OX to XY to get OY.

A scalar is just a number.

• Multiplying a vector with a scalar obviously gives you a different vector.
• Multiplying a vector by another vector gives you a scalar.

The scalar product of two vectors comes from multiplying them together like node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , so called because you get a scalar out of it. Different from Cross product. If the scalar product is zero, the vectors are orthogonal to each other. The algorithm is as follows (where φ is the angle between the vectors):

node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

The second clause is necessary because, though node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , any angle φ between a vector and the null vector is undefined, breaking the math.

You can see that if the angle is 90 degrees – i.e. they are orthogonal – the scalar product will be zero, because what is the cosine of 90 degrees?

Note, often in linear algebra, the angle between two objects indicates the smaller of the two possible angles – the pointy angle, so node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 .

A formula that does not use the angle, is this. Let node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , and node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 . Then

node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

So named because a you get a vector out of it. Also called cross product. Only defined in 3D-space.

There is no "the" normal-vector of a plane. A normal-vector is determined by its direction, but its length can be whatever you like. As such, the cross product of two vectors within a plane is one possible normal to that plane.

To get the coordinates of a vector product, the practical method is this:

Let node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 and node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 . (Coordinates are given in a fixed ON-base, this time positively oriented). Then

node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

As a memory rule, you can use the nonsensical expression

node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

nonsensical because determinants shouldn't contain vectors, but it works here as a memory rule to produce the formula. Do not forget the negative sign in the formula.

The magnitude, or length of a vector, is calculated with the Pythagorean theorem.

Want to find the parallellogram between two vectors? Take the cross product of the two, and calculate the absolute value of it. It's the same as the determinant of the two.

In two-dimensional space, two vectors u and v, create a positively oriented base if:

when you rotate the first vector, u, counter-clockwise, you can make it point in the same direction as v without rotating more than 180 degrees.

In three-dimensional space… imagine yourself perched on the top of the third vector, looking down at the plane made by the other two. Apply the above rules.

(VERIFY) Alternatively, you can use the hand rule:

Form your right hand so that your thumb, index and middle fingers are orthogonal to each other. May require some stretching of muscles.

You can pivot your hand any way you like, the point being… if your palm is facing upwards, your middle finger is pointing upwards, and the thumb to your right. Then the thumb should represent the x-axis (the first vector), the index finger the y-axis (the second vector), and the middle finger the z-axis (or a cross product of two vectors). Now that you've named your fingers, you can pivot the hand around.

If the vectors you are looking at don't follow that arrangement, they create a negatively oriented base(?).

An orthonormal base is one where the base vectors are orthogonal to each other and have length 1.

There are several different ON-bases: it can be orientated all manner of ways. In particular, some rules are different if it has a "positive" versus "negative" orientation, see Positive/negative orientation.

If in R³ you have a plane node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , which can also be written node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , how does the plane look?

Think before you read on!

…

Does it matter what y or z are?

The plane doesn't give a shit about y or z, but x must be zero. So, at any point on the plane, x must be zero. That means, that the x-axis is orthogonal to our plane.

Good going! Now, find a normal to that plane. It is easy: pick any value of x: node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , for instance.

Notice how you can get that from reading the plane's equation: they are the coefficients to x, y, z.

You know that it is a normal because the Scalar product of orthogonal vectors should be zero, right? A vector node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 within the plane multiplied with node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 is zero. So an alternative way to express the equation of that plane is node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 . IOW by writing this, we are considering all vectors node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 whose dot product with node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 is 0, and they all lie within that plane.

Let a new plane node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 . You can express this as node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , or node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , so it is a plane parallel to node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 but that is -128 units away from it in the x direction.

Let a new plane node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 . Once again, you can find a normal by taking the coefficients: node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , because the value of node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 remains constant when you move perpendicularly to node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , by linearity and definition of perpendicularity.

Use both ways to do the dot product. The only unknown becomes node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , which you can solve for.

Also, a trick is: the angle between two lines is the same as the angle between their normals. (Draw this)

To project the vector u on the vector v, the formula is

node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

Look at R². If you have two vectors pointing different directions, you can create any other vector in R² by scaling those two and adding them to each other an appropriate amount of times.

If those two vectors are on the same line, you can only ever get another vector on that line. Make sense?

Zero vectors, regardless of scaling, can only produce another zero vector.

Now, you can extend this to higher dimensions. If two vectors are within a plane, there is no combination of those two vectors that can produce a vector poking out of the plane. It is said that any set of three vectors within the plane is a linearly dependent set, because the other two can be used to produce the third – there is redundancy.

A linearly independent set of three vectors within R³ is one where none of the vectors can be produced by a combination of the other two.

To check for linear dependence, e.g. to check if three vectors within space are in the same plane, put them in a determinant like so: det(u, v, w):

node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 If the determinant is zero, the vectors are linearly dependent.

determinant=0, system of equations solutions, invertability of matrix

• Math

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