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As a super-important example, the volume of a parallelepiped is givenThis misses a vitally important piece of information. Trivector A^B^C is more than just volume (and shape) of the respective parallelepiped. It also decrees the way we go around the parallelepiped along its edges.
by the trivector A∧B∧C which is equal to A×B·C ... definitely not A×B×C.
All we need is a sense of circulation around the edge.But this is in conflict with:
The faces can be indistinguishable. They can be transparent.Marking two faces differently is equivalent to distinguishing between two opposite senses of circulation around the edge. The reason is that a bivector is not just a parallelogram made of segments, but a parallelogram made of directed segments (vectors!) and in a way (tail to tip!) creating the sense of rotation. Therefore making the faces transparent would only mask the existing handedness (chirality), but not eliminate it. Transparent sides will not eliminate the difference between A^B and B^A! Generally, I find the table:
scalar point - no geometric extent; grade=0.>vector-length-geometric extent in 1 direction; grade=1>bivector-area-geometric extent in 2 directions; grade=2>trivector-volume-geometric extent in 3 directions; grade=3, etc.correct only in its first line. The rest is misleading, because vector is not just the length, it is directed segment, etc.
Reference [1]: D. Hestenes, Am. J. Phys., 71 (2), 2003
There's something wrong with that reference. I assume it refers to>David Hestenes