Below, we'll need a consequence of the ROTATION OPERATOR, R -- usually formulated as a matrix in cos q, sin q.
I'm DERIVING the (EUCLIDEAN) INNER PRODUCT (a.k.a. SCALAR PRODUCT) OF VECTORS, familiar in The Literature. Typically, THE LAW OF COSINES is DERIVED FROM INNER PRODUCT. But I, UNIQUELY, reverse this procedure --DERIVING by two MOTIVES:
- TRANSFORMATION OF A ("SCALAR") UNDIRECTED TRIANGLE TO A DIRECTED TRIANGLE (TRIANGLE OF VECTORS);
- CORRESPONDING TRANSFORMATION OF LAW OF COSINES.
Given a minimum of analytic or coordinate geometry (and associated algebra), Augustin Cauchy (1789-1857) taught us how to derive all of trigonometry.
Let x = q. Then, x2 + y2 = 1 → cos2 q + sin2 q = 1.
We wish to interpret [2] in terms of [1], in TRANSFORMING AN UNDIRECTED D INTO A DIRECTED D.
- SQUARES in [1] suggest SQUARING [2].
- But cos C = cos q in [1] suggests more than a "polynomial product".
- Let us propose a "vector product", denoted ·, such as a·a.
- Then we have (from [2]):
c·c = (b — a)·(b — a) → c·c = b·b — a·b — b·a + a·a. [3]- We can now MATCH [3] WITH [1] (TRANSFORMING COSINE LAW INTO A NEW PRODUCT) BY:
- DECLARING · is COMMUTATIVE, with a·b = b·a, so we can COLLECT ¯(2a·b) in [3];
- DECLARING a·b = |a| |b| cos q, q= Ð(a,b) to match corresponding terms in [1], [3];
- DECLARING vector inner product to be scalar since
- this is so by the SV Principle for self inner products, wherein q = 0;
- and CLOSURE is obtained by DECLARING IT FOR ALL INNER PRODUCTS.
ASSIGNMENT: Improve REASONING and LANGUAGE of this DERIVATION.
What's an IDEMPOTENT? Wots it gud fer? AN IDEMPOTENT (in ARITHMETIC or ALGEBRA) is an ELEMENT WHOSE SQUARE EQUALS ITSELF. You already know two IDEMPOTENTS in ARITHMETIC: 0 and 1. Thus 0 · 0 = 0; and 1 · 1 = 1. 0 is THE IDENTITY ELEMENT OF ADDITION (AND THE ADDITIVE GROUP); 1 plays those roles for MULTIPLICATION (and THE MULTIPLICATIVE GROUP). Let's say these are "the improper idempotents". Let's construct a PROPER IDEMPOTENT, then explain its use.
(a + bd3) · (a + bd3) = a2 + 2abd3 + b2 º a + bd3 → (a2 + b2) + 2abd3 º a + bd3.
The vector term simplifies by dividing "out" bd3 on both sides: 2a = 1 -> a = 1/2.
INNER PRODUCT has a special property which tells us something about OUTER PRODUCT and MULTIPRODUCT.
The ROTATION OPERATOR, R, is a SPECIAL CASE of AN ORTHOGONAL TRANSFORMATION, PRESERVING SCALE.
- Let O denote AN OPERATOR;
- appeal to the Einstein convention whereby paired subscripts indicate summation;
- apply the Kronecker symbol (denoted "S", rather standard d used for BASIS UNIT), Sij = +1, if i = j, otherwise 0;
- then OijOjk = OjiOki = Sij. [4]
TRANSFORMATION INVARIANT: A SCALAR PRESERVED BY A SCALE-PRESERVING TRANSFORMATION.
Hence, INNER PRODUCT IS A TRANSFORMATION INVARIANT:
a · b = aibi = a'ib'i = OijOikajbk = Sijajbk = akbk. [5]
This INVOKES THE DEFINITIVE CHARACTERISTIC OF EUCLIDEAN SPACE. For (as Henri Cartan notes) a point in n-dimensional Euclidean space can be defined as a set of numbers (coordinates), [x1, x2, ..., xn], such that the distance of this point [x] to the ORIGIN, namely, [0,0,...,0] -- or, equivalently, the self inner product x x of the vector x, from point x to point 0 -- is given by THE FUNDAMENTAL FORM:
x = x21 + x22 + ... + x2n = x · x. [6].But pseudo-Euclidean spaces (as required, say, in relativistic space-time) may have a DIFFERENT FUNDAMENTAL FORM:
x = x21 + x22 + ... + x2n—k — x2n—k+1 — ... — x2n. [7]Clearly, [7] cannot be CONSTRUCTED BY INNER PRODUCT. So, ANOTHER TYPE OR VECTOR PRODUCT IS REQUIRED. MY UNIQUE DERIVATION OF OUTERPRODUCT.