What is the inverse of a vector?

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As a relatively recent personal example I spent a few months (in bits and pieces) working out a bunch of metrical spherical geometry for myself without reference to past work, with points represented as displacement vectors to stereographically projected points at https://observablehq.com/@jrus/planisphere with the eventual goal of implementing computational geometry / cartography code using that as a canonical representation, which I think is superior to representations used currently in practical software.

The same spherical relationships are comparable (some things slightly slightly easier, some slightly trickier) to represent as displacement vectors in an embedded sphere. But there again relationships are clearer to express in GA terms.

Most of the material there is stated without proof (maybe eventually full proofs should be included), but several of the identities there I worked out very tediously with pages of scratch work in coordinates, then realized afterward the same results could be arrived at with only a few lines of GA.

Only a bit of the material is truly novel (after doing the work myself, I hunted around for sources and found some of the same formulas worked out previously using classical spherical trigonometry 200+ years ago), and e.g. some very similar material where the stereographically projected points are represented as complex numbers can be found at http://fer3.com/arc/img/110279.applications%20of%20complex%2...

In theory most of the rest could be also worked out using complex numbers or matrices, but (a) some ideas end up awkward and unidiomatic there so you would never think to do it, so that many identities that are slightly obscure in GA are almost unheard of written in other formalisms, (b) the algebraic manipulation is at least 2–3x more cumbersome.

That meant you could take AO (a halfway house between O and A) when everyone else was taking their normal O levels. The thing is that the extra stuff in AO was all Pure Maths, and formed a fair amount of the easier "P1" maths syllabus for the normal A level maths exam, which had P1 and Me1 (Maths with mechanics 1, basically statics).

Because you'd done that work already prior to the A level years, you could take "A level maths" after only 1 year (which looked really good on UCCA applications :), so you've now done an exam consisting of the two 'P1' and 'Me1' papers in the first year of your A levels.

Which meant that in the second year of your 'A' levels, you could do 'Pure Maths' (P1, P2) and 'Further Maths' (Me1, Me2) for a total of 3 maths A levels.

On top of that, you had your other two subjects (mine were Physics and Chemistry), and because it was the JMB board, everyone got to do "General Studies".

Getting all of them gave you 6 A levels, even though some of the work was duplicated in the maths arena (over different years of course :)

S levels were a bonus on top - there was no fudging for those, though, you just took what you thought would be useful to study. They gave me maths and physics because I'd said I was going to do physics at college... :)

The distinct gets a little more useful when you start dealing with covariance under coordinate transformations. There it becomes more meaningful, because the _vector_ given by r x F doesn't transform the same way as their cross product should.

For an obvious example of why this is true: suppose r=x and F=y. Then r × F = z. If you change coordinates by mapping z -> 2z, then you would be doubling the torque that you computed .. which is wrong; the torque is unchanged. The bivector x^y is correctly unchanged by z -> 2z.

Currently in physics courses (usually not until more advanced mechanics or relativity) the resolution to this is to wave ones' hands and declare that, no, torque is a 'pseudovector'. But it is really much easier to think about if you type it as a bivector in the first place.

Heh, that reminds me of my first physics course in an under-graduate computing degree (in Romania). The curriculum was so well designed overall that this Physics course needed linear algebra concepts that would be taught halfway-through the semester in Algebra, integration along a surface and similar that would be taught at a similar in the Calculus course, and some Statistics I don't remember that would be only be taught in the second semester.

The prof's solution? He taught a 3-hour course covering all of the above, and considered that good enough for all future courses. This particular Physics course later went on to cover analytical mechanics (generalized coordinates, Lagrangians, Hamiltonians), electricity, general relativity, statistical thermodynamics, and quantum mechanics, all in a single semester.

Needless to say, 99% were happy they passed and couldn't tell you a single thing about any of these subjects a few minutes after the final exam.

Any vector space over a field (usually part of the definition of a vector space) is equipped with the standard inner product, because multiplication and addition are part of the definition of a field.