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# Pseudovectors

To get to the other side of the block, should you go ‘right and then left’ or ‘left and then right’? It of course doesn’t matter. ‘$20 \,\mathrm m$ east and then $30 \,\mathrm m$ north’, or ‘$30 \,\mathrm m$ north and then $20 \,\mathrm m$ east’ – the result is the same. Let’s call this a commutative law1 for addition2

$$\vec s_\text{path1}+\vec s_\text{path2} =\vec s_\text{path2} +\vec s_\text{path1}$$

But, repeat for angular position vectors $\vec \theta$ and suddenly the order does matter!

$$\vec \theta_\text{up}+ \vec \theta_\text{around}\neq \vec \theta_\text{around}+ \vec \theta_\text{up}$$

The end-result is different if the order of actions is different. Angular vectors3 seem to have a commutative-law issue. In fact, any vector that does not point the same way as its property acts,4 has this issue.

• Let’s call those that do obey the commutative law geometric vectors.5 6
• When the commutative law is not obeyed, that’s when we call it a pseudovector.7

Now, why is there this difference? It turns out to have something to do with symmetry. When we invent pseudovectors, we “mess with” the natural symmetry. For instance, think of a bike driving forward. Imagine that you mirror this situation, so the biker’s left side becomes his right side. All geometry is flipped, and all properties should flip as well so it all still fits. But clearly the angular velocity does not flip! The right-hand rule8 still causes it to point leftwards and it didn’t flip to rightwards.

This is an example of broken symmetry causing the issue.9

References:

1. Online Etymology Dictionary’ (dictionary), Douglas Harper, www.etymonline.com
2. euclidian vector’ (web page), Definitions.net, STANDS4 LLC, 2020, www.definitions.net/definition/euclidean+vector (accessed Jan. 30st, 2020)
3. Euclid’ (encyclopedia), N. S. Palmer, Ancient History Encyclopedia, 2015, www.ancient.eu/Euclid/ (accessed Jan. 30st, 2020)
4. Geometriske vektorer’ (article, in Danish), Karsten Schmidt, The Technical University of Denmark, Institute for Mathematics and Computer Science, eNote 10, 2016, 01005.compute.dtu.dk/enotes/10_-_Geometriske_vektorer (accessed Jan. 30st, 2020), chapter 10
5. Geometric Transformations’ (web page), Michigan Technological University (MTU), pages.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry/geo-tran.html (accessed Mar. 12th, 2020)
6. Maths - Reflection’ (web page), Martin John Baker, Euclidean Space, 2007, www.euclideanspace.com/maths/geometry/affine/reflection/index.htm (accessed Mar. 12th, 2020)
7. Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics’ (book), David Hestenes, Springer, 1987, www.amazon.co.uk/Clifford-Algebra-Geometric-Calculus-Mathematics/dp/9027725616, ISBN 978-9027725615
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