Re:What is the inverse of a vector?
by tgb on Wednesday November 03, 2021 @07:04AM
(#29093067)
Yes, only three. As defined, two bivectors are equal if their areas are equal and if their oriented planes are equal. Therefore two more degrees of freedom are absorbed by taking rotations of the two vectors in the plane.Along with the rescaling the author noted, we're down to three from six.
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That makes complete sense to me. But then later on they say "The output is a Geometric with a scalar component s and a bivector component ⇒c, which has 1 + 5 = 6 degrees of freedom so this system is not lossy! It should permit an unambiguous inversion operation!" If a bivector only has 3 degrees of freedom then the total is 4, which seems like it would be lossy?
I was also wondering this. But note that x^y is always perpendicular to x, so really only has two degrees of freedom while you need three to recover y (knowing x). Add in the dot product part to make up for it.
Re:What is the inverse of a vector?
by kevinwang on Wednesday November 03, 2021 @09:21AM
(#29094209)
Interesting, I was confused about the same thing. So the author is not correct when they say a bivector has 5 degrees of freedom?
Re:What is the inverse of a vector?
by mferraro89 on Wednesday November 03, 2021 @12:37PM
(#29097001)
Yes, a bivector has 3 degrees of freedom. I was mistaken when I wrote 5. I'll fix the text tonight.