RESULTS

RESULTS FOLLOWING FROM INTRODUCTION OF OUTER PRODUCT
Given vectors, ab induces an ORIENTED PARALLELOGRAM in a PLANE. So, we find: |ab| = |a| |b| sin C. [1]
Given this, please recall the vector triangle used in DERIVING INNER PRODUCT FROM THE LAW OF COSINES, namely,a + b = c. [2]
Forming outer product with vector a or b or c in [2], we have: aa + ab = ac → ab = ac[2a]|| ba + bb = bc → ba = bc[2b]|| ca + cb = cc → ca = cb = 0.[2c]
From [2a], [2b], [2c], we have: ab = ac = bc. [2d]
Adapting [1] to [2d], we have: ab = |a| |b| sin C = ac = |a| |c| sin B = bc = |b| |c| sin A → (sin C)/c = (sin B)/b = (sin A)/a. [3]
We've DERIVED THE LAW OF SINES by APPLYING OUTER PRODUCT TO A TRIANGLE. And, in THE LITERATURE, we find DERIVATION of THE LAW OF COSINES by APPLYING INNER PRODUCT TO A TRIANGLE. This and the following leads to an insight.
Russian mathematicians sometimes make DIMENSIONAL DISTINCTIONS REGARDING TRIANGLES (OR POLYGONS): THE 0-TRIANGLE CONSISTS MERELY OF THE VERTICES OF THE TRIANGLE; THE 1-TRIANGLE CONSISTS OF THE LINE SEGMENTS COMPOSING THE TRIANGLE; THE 2-TRIANGLE IS THE 1-TRIANGLE FILLED IN.
We may say that THE LAW OF SINES is A STATEMENT ABOUT THE 1-TRIANGLE and THE LAW OF COSINES is A STATEMENT ABOUT THE 2-TRIANGLE.
Hey! There's another connection here.

In passing from an undirected geometric structure to a directed one, the Law of Cosines leads to (connects with) te vector INNER PRODUCT.
By a similar passage, the OUTER PRODUCT connects with the Law of Sines
.
A BASIS VECTOR, such as d₁ = [1, 0, 0], denotes A DIRECTED LINE SEGMENT. Its OUTER PRODUCT WITH another BASIS VECTOR, such as d₂ = [0, 1, 0], namely, d₁ d₂, denotes AN ORIENTED PLANE SEGMENT (here, A SQUARE). The OUTER PRODUCT of THE PREVIOUS OUTER PRODUCT WITH BASIS VECTOR, such as d₃ = [0, 0, 1], namely, d₁d₂d₃, denotes A DIRECTED 3D-SECTION (here, A UNIT CUBE). From such notions, the geometric and vector properties can be built up.
OUTER PRODUCT explains something occasionally taught as a curiosity (without explanation) in analytic geometry. Draw a triangle in the Cartesian Plane. Construct a DETERMINANT from the triangle's vertex coordinates: row i = (x_i, y_i, 1), i = 1, 2, 3 -- a listing of vertices as triangle is rotated counterclockwise. Then, THE AREA OF THE TRIANGLE IS ONE-HALF THE DETERMINANTAL VALUE. However, if vertices are listed clockwise, or any two rows of the above ordering are interchanged, then the above VALUE is multiplied by negative one. This shows that AREA IS AN ORIENTED STRUCTURE.

MULTIVECTORS under OUTER PRODUCT are EQUIVALENT to ANTISYMMETRIC TENSORS. Thus, given BASIS VECTORS, d₁ and d₂, and BIVECTOR, b, WE CAN CONSTRUCT a (contravariant) SECOND-ORDER TENSOR: B_ij = ¯(d₁d₂) ÄB. Given also d₃ and a TRIVECTOR, T, we can construct a (contravariant) THIRD-ORDER TENSOR: T_ijk = ¯(d₁d₂d₃)ÄT.
Symmetric forms, familiar in NUMALGEBRA, are generated by COMMUTATIVITY; alternating forms, as in MULTIVECTOR THEORY beginning with OUTER PRODUCT, are generated by ANTICOMMUTATIVITY. Students can be guided to link these forms back to the ALGORITHMS of "casting out nines" and "casting out elevens". Decimally notated number, d = 18117 has decimally notated number 9 as a FACTOR because the SUM OF DIGITS in d -- namely, 1 + 8 + 1 + 1 + 7 = 18, and 1 + 8 = 9 -- YIELDS ZERO UPON CASTING OUT ANY NINES. Also, d has decimally notated number 11 as a FACTOR because THE ALTERNATING SUM OF DIGITS IN d -- namely, reading from the right, 7 - 1 + 1 - 8 + 1 = 0 -- YIELDING ZERO WHEN ELEVENS ARE CAST OUT. These are, respectively, the properties of "easy detection of decimal-base-minus-one and decimal-base-plus-one". These properties CARRY OVER to POLYNOMIALS in "the unknown-base-"x for "easy detection of FACTORS (x - 1), (x + 1) -- that is, "unknown- base-minus-one, unknown-base-plus-one". Thus, the polynomial, 2x⁴ — 3x³ + 2x² + 3x — 4, has FACTOR, (x — 1), because THE SUM OF ITS COEFFICIENTS IS ZERO: 2 — 3 + 2 + 3 — 4 = 0. This polynomial also has FACTOR (x + 1) because ITS ALTERNATING SUM IS ZERO: (beginning on right) — 4 — 3 + 2 + 3 + 2 = 0. Hence, we have the following homology:
nines-rule : elevens-rule :: (x - 1)-rule : (x + 1)-rule :: symmetric forms : alternating forms.
More results? (Warning-on-the-Label: Unless you're a mathematician or math student, this file may provoke christopher-robinitis, a.k.a., THE SNEEZLES.)