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- URL: https://x.com/EricLengyel/status/2019901780549861535
- Author: Eric Lengyel (@EricLengyel)
- Posted: 2026-02-06 22:31:27
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1/ @EricLengyel
When a is a vector and b is any multivector, the geometric product a ⟑ b decomposes as
a ⟑ b = a ∧ b + b ∨ (a
and b ⟑ a decomposes as
b ⟑ a = b ∧ a + (a_★) ∨ b.
If a and b are both vectors, these both reduce to
a ⟑ b = a ∧ b + a ⋅ b.
Branches: 2026-02-07-NateMorrical-so-in-the-case-of-two-vectors-the-result-of-the
2/ @EricLengyel
Here, ^★ and _★ are the right and left Hodge duals.
The expression b ∨ (a^★) is the definition of the right contraction, sometimes written b ⌊ a. Likewise, (a_★) ∨ b is the left contraction a ⌋ b.
Branches: 2026-02-06-pedroth9-never-could-understand-right-or-left-contraction