I hate it when mathrmaticians or physicist start with the "interesting" properties. It usually means i am going to get lost!
Outer product is the exterior product and is related to the exterior angle of a convex figure. You get an exterior angle by producing one side relative to its adjoint neighbour. the angle measure produce is say θ then the outer product for sides a,b is absinθ.
The dot product is called the inner product and it represents a projection of the adjoint neighbour onto the produced part of the relative side. it basically is a measure of how independent that side is of the relative side, which is a measure of how parallel . The inner product for the sides a,b is abcosθ
This is fine for the plane containing the 2 basis vectors or reference orientations for measurement, and particulrly if those orientations are perpendicular. The plane by the way could have any orientation in 3d space.
We have a way of dealing with generalised coordinate frames in the plane by defining a transformation from perpendicular to angled vectors and defining thes vectors, in terms of the perpendicular reference frames as covariant and contravariant. Essentially what we do is calculate the parallax and adjust the vector scalars accordingly. We make the assumption that the length of an object is invariant in the 2 reference frames.
Because the reference frame angles arestill measured using the perpendicular frame, even though the measuring frame for distance is not perpendicular,we can do this and we get invariance in the definition of the inner and outer products.
Now suppose i use a set of 3 orientations to measure in the plane, and for convenience i rotate them equally so they are symmetrical rotationally, what then?
Well the first thing to realise is that i ave a 3 dimensional space.
It is flat but it has 3 dimensions. The second thing is we will not use a perpendicular reference frame to resolve this.
What can we do?
Well in fact Abraham De Moivre solved this little conundrum by using imaginary magnitudes, in the time of Newton, and it is Called the De Moivre Theorem or more fully the Cotes De Moivre theorem. It is usually explained in terms of the roots of unity, but since grassmann, and particularly Peanos version of Grassmann it has been looked at in terms of polysigns, a generalization of the notion of signs ostensibly,
As interesting as these different names may sound it must not be aallowed to cause confusion or division any longer.
Grassmann dealt with the issue of ndimensional spaces and subspaces and gave the rules which apply geometricall to any case, but he had to accept the rules of sign.
The rules of sign sounds grand, and it is easy to go off on an interesting tangent relating it to complec numbers, or as they ued to be called imaginaries. In fact it is none of these.
After years of research i have concluded that a few simple mistakes, misunderstandings, humman pride and self aggrandizement have contributed to the wholly inadequate "number line" conception, which became the number set conception which then extended number into the basis of all topological descriptions of space in the most complex and abstract way.
The mistake came through the hared of the negative integers, and through the misreading of Arithmoi throuhout Europe. Within Arabia and Persia, where the Euclid documents were freely available o scholars along with the Ptolemaic Almegest the understanding of ratio and proporton was absolute, and trigonometry and The trigonometric ratios ere a topic of intense and serious study and refinement. Here Brahmagupta was less misunderstood than in the west.
The rules of signs belong to the extension of trigonometry to spherical trigonometry. The familiarity of geometers with these issues lead to many relationships between the ratios, which enabled the most suitab;e relation to be picked for any situation. In particular when the angle of arc exceeded π/2 radians tables were available to calculate the necessary values.
The use of the Almegest modified according to the indian practice taught by Brahmagupta, not only facilitated calculation but revealed deep relations between the right gnomon, the arec of angle, and the radius of the enclosing circle. These reltionships were written in so many different ways and with so many different names that you needed to be expert to follow what was being utilised. For example, the sine was the main ratio, Then came the haversine, the versine the secant and the tangent . Eventually the cosine and the invers trig ratios, and then the secant and cotangent.
The spheical trigonometrist and geometrist a persian suggested the use of a scalar notation to represent these ratios. His suggestion was not fully implemented until Regiomantus who introduced the Fractional notation or fractions. These were later renamed rational numbers, because it was understood they represented ratios. They were called scalars because that represented the proportions required to change from one scale to another.
Brahmaguptas misfortunate numbers were designed for this situation, and in the tradition of vedic sutras facilitated calculation. As a consequence it soon became clear that many ratios could be written in terms of each other providing one used and understood brahmaguptas misfortunate numbers.
Very few liked misfortunate numbers, partly due to the name , partly due to the astrilogical significance, and partly due to the unfamiliarity with rapid calculation techniques the indians excelled in, but mostly due to the religious oppression of the Roman church in the west. The Islamic reformation was denounced as the work of the devil, and books on alchemy and algebra were burnt along with the Q'uran and the owner of the books. Truely these numbers were misfortunate or anyone who knew or utilised them.
Brahmagupta was innovative, but a bit of a fuddy duddy and he did not do a great jobon explaining his misfortunate numbers to the arabic and eastern communities. It was universally believed and taught tha nothing could come from nothing, which was why god was supposed to be so incredible, creating "ex nihillo". So when brahma gupta seemed to be saying you can think of these ciphers as being cut grom "nothing", he ran into disbelief, confusion and hostility!
Actually what he said is you can think of them as being cut from shunya. This is entirely different and contrasts with the normal adding of things to shunya. Shunya is the fullness of space.
Thus the notation of quadrant or gnomonic position wasidden in the angle maurealgorithms. The inclusion of misfortunate numbers was rationalised to the notational convention of sign, which in major part was due to Bombelli drawing on Cardano and Taglietelli(!) and others. But it was not until Vieta and the later german school of notation makers made the familiar + and – standard.
However, the twist in Europe was the horrendous misconception √-1. Bombelli solved it in a practical way. Newton and Cotes and De Moivre integrated it into the then mainstream of Mathematics, where its relationship to the Trigonometric functions was proclaimed, fully explored, but not accepted until much later, despite Cotes deriving the logarithmic equivalent of Eulers equation, and thus the first true vector equation of import.
Hamilton provided undeniable logical justification for its meaning and relevance. However, the work started by Bombelli, Based securely on Euclidean geometry was derailed until Grassman, Who despite the failing, was able to continue to derive a Euclidean geometric calculus.
The sign rules, put forward by Bombelli are eventually generalised into the roots of unity allowing n dimensional space to be properly aggregated. Thus these are nothing more than the trigonometric sign rules introduced by Brahmagupta, replaced into the right position in geometry and revealing the many spaces in perpetual motion that Brahmagupta envisaged.
a'a*sin (2π/k*θ) is the exterior product for any 2 adjacent vectors a' and a in a k dimensiona space θ being a real number. If θ is measured only between the 2 vectors then it takes a different rational form allowing the product to vanish when parallel to any vector spanning the space. The sign of the trig ratio then comes to the fore necessitating explicit treatment to ensure that the algorithm functions correctly.