kend
April 21, 2020, 8:14pm
#1
Hello,

I am new to geometric algebra and saw a comment on another post stating that the square of an orthogonal basis vector belongs to the set {-1, 0, +1}.

Is there a mathematical reason why e_{i} ^{2} have to map to an element in the set {-1, 0, +1} or is just a convention that proves to have utility?

jrus
April 22, 2020, 1:50am
#2
Basis vectors donâ€™t need to be unit length or orthogonal to each-other. But if they arenâ€™t, you can then pick a new basis for the same space which does satisfy those constraints, and is usually more convenient to work with.

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Hi @kend ,

you can get a look at the algebra \mathbb{R}^{4,4} of Goldman and Mann. The basis vectors still square to 0 but the metric is full of 0.5 :

\begin{array}{cccccccc}
0 & 0 & 0 & 0 & 0.5& 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.5 &0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0.5 &0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5\\
0.5 & 0 & 0& 0 & 0 & 0 & 0 & 0\\
0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.5& 0 & 0& 0 & 0 & 0\\
0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0\\
\end{array}

They could have set only â€ś1â€ť instead of â€ś0.5â€ť, this choice is motivated to avoid power of 2 in the versors.

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