Vector Geometric Algebra

\(G_{3,0,0}\) using right-hand convention.

NOTE: \(G_{3,0,0}\) describes a structure with 3 basis vectors whose squares are 1.

Algebraic Structure

Basis vectors and blades:

\(e\)

\(e_1\)

\(e_2\)

\(e_3\)

\(e_{12}\)

\(e_{23}\)

\(e_{31}\)

\(e_{123}\)

\(1\)

\(\mathbf{x}\)

\(\mathbf{y}\)

\(\mathbf{z}\)

\(\mathbf{xy}\)

\(\mathbf{yz}\)

\(\mathbf{zx}\)

\(\mathbf{xyz}\)

Square table:

\(e^2\)

\(e_1^2\)

\(e_2^2\)

\(e_3^2\)

\(e_{12}^2\)

\(e_{23}^2\)

\(e_{31}^2\)

\(e_{123}^2\)

\(1^2\)

\(\mathbf{x}^2\)

\(\mathbf{y}^2\)

\(\mathbf{z}^2\)

\((\mathbf{xy})^2\)

\((\mathbf{yz})^2\)

\((\mathbf{zx})^2\)

\((\mathbf{xyz})^2\)

\(1\)

\(1\)

\(1\)

\(1\)

\(-1\)

\(-1\)

\(-1\)

\(-1\)

Cayley Tables

Wedge product:

\(\wedge\)

\(1\)

\(\mathbf{x}\)

\(\mathbf{y}\)

\(\mathbf{z}\)

\(\mathbf{xy}\)

\(\mathbf{yz}\)

\(\mathbf{zx}\)

\(\mathbf{xyz}\)

\(1\)

\(1\)

\(\mathbf{x}\)

\(\mathbf{y}\)

\(\mathbf{z}\)

\(\mathbf{xy}\)

\(\mathbf{yz}\)

\(\mathbf{zx}\)

\(\mathbf{xyz}\)

\(\mathbf{x}\)

\(\mathbf{x}\)

\(0\)

\(\mathbf{xy}\)

\(-\mathbf{zx}\)

\(0\)

\(\mathbf{xyz}\)

\(0\)

\(0\)

\(\mathbf{y}\)

\(\mathbf{y}\)

\(-\mathbf{xy}\)

\(0\)

\(\mathbf{yz}\)

\(0\)

\(0\)

\(\mathbf{xyz}\)

\(0\)

\(\mathbf{z}\)

\(\mathbf{z}\)

\(\mathbf{zx}\)

\(-\mathbf{yz}\)

\(0\)

\(\mathbf{xyz}\)

\(0\)

\(0\)

\(0\)

\(\mathbf{xy}\)

\(\mathbf{xy}\)

\(0\)

\(0\)

\(\mathbf{xyz}\)

\(0\)

\(0\)

\(0\)

\(0\)

\(\mathbf{yz}\)

\(\mathbf{yz}\)

\(\mathbf{xyz}\)

\(0\)

\(0\)

\(0\)

\(0\)

\(0\)

\(0\)

\(\mathbf{zx}\)

\(\mathbf{zx}\)

\(0\)

\(\mathbf{xyz}\)

\(0\)

\(0\)

\(0\)

\(0\)

\(0\)

\(\mathbf{xyz}\)

\(\mathbf{xyz}\)

\(0\)

\(0\)

\(0\)

\(0\)

\(0\)

\(0\)

\(0\)

Dot product:

\(\cdot\)

\(1\)

\(\mathbf{x}\)

\(\mathbf{y}\)

\(\mathbf{z}\)

\(\mathbf{xy}\)

\(\mathbf{yz}\)

\(\mathbf{zx}\)

\(\mathbf{xyz}\)

\(1\)

\(1\)

\(\mathbf{x}\)

\(\mathbf{y}\)

\(\mathbf{z}\)

\(\mathbf{xy}\)

\(\mathbf{yz}\)

\(\mathbf{zx}\)

\(\mathbf{xyz}\)

\(\mathbf{x}\)

\(\mathbf{x}\)

\(1\)

\(0\)

\(0\)

\(\mathbf{y}\)

\(0\)

\(-\mathbf{z}\)

\(\mathbf{yz}\)

\(\mathbf{y}\)

\(\mathbf{y}\)

\(0\)

\(1\)

\(0\)

\(-\mathbf{x}\)

\(\mathbf{z}\)

\(0\)

\(\mathbf{zx}\)

\(\mathbf{z}\)

\(\mathbf{z}\)

\(0\)

\(0\)

\(1\)

\(0\)

\(-\mathbf{y}\)

\(\mathbf{x}\)

\(\mathbf{xy}\)

\(\mathbf{xy}\)

\(\mathbf{xy}\)

\(-\mathbf{y}\)

\(\mathbf{x}\)

\(0\)

\(-1\)

\(0\)

\(0\)

\(-\mathbf{z}\)

\(\mathbf{yz}\)

\(\mathbf{yz}\)

\(0\)

\(-\mathbf{z}\)

\(\mathbf{y}\)

\(0\)

\(-1\)

\(0\)

\(-\mathbf{x}\)

\(\mathbf{zx}\)

\(\mathbf{zx}\)

\(\mathbf{z}\)

\(0\)

\(-\mathbf{x}\)

\(0\)

\(0\)

\(-1\)

\(-\mathbf{y}\)

\(\mathbf{xyz}\)

\(\mathbf{xyz}\)

\(\mathbf{yz}\)

\(\mathbf{zx}\)

\(\mathbf{xy}\)

\(-\mathbf{z}\)

\(-\mathbf{x}\)

\(-\mathbf{y}\)

\(-1\)