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Vector Addition
Combines two vectors using the parallelogram rule. The result is
the diagonal of the parallelogram formed by the two vectors.
C = A + B
Grade: 1 (vector)
Commutative: Yes
Associative: Yes
Commutative: Yes
Associative: Yes
Mathematical Background
Geometric Algebra extends vector algebra with blades of different grades:
- Grade 0 (Scalar): Pure numbers, magnitude without direction
- Grade 1 (Vector): Directed lengths, e₁ and e₂ basis
- Grade 2 (Bivector): Directed areas/rotations, e₁∧e₂ basis
- Grade 3 (Trivector): Directed volumes (in 3D)
The Geometric Product combines inner (dot) and outer (wedge) products:
ab = a·b + a∧b
Rotors generalize quaternions: R = exp(-θB/2) where B is a bivector