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:
\(\cdot\) |
\(1\) |
\(\mathbf{x}\) |
\(\mathbf{y}\) |
\(\mathbf{z}\) |
\(\mathbf{xy}\) |
\(\mathbf{yz}\) |
\(\mathbf{zx}\) |
\(\mathbf{xyz}\) |
---|---|---|---|---|---|---|---|---|
\(1\) |
\(1\) |
\(\mathbf{x}\) |
\(\mathbf{y}\) |
\(\mathbf{z}\) |
\(\mathbf{xy}\) |
\(\mathbf{yz}\) |
\(\mathbf{zx}\) |
\(\mathbf{xyz}\) |
\(\mathbf{x}\) |
\(\mathbf{x}\) |
\(1\) |
\(0\) |
\(0\) |
\(\mathbf{y}\) |
\(0\) |
\(-\mathbf{z}\) |
\(\mathbf{yz}\) |
\(\mathbf{y}\) |
\(\mathbf{y}\) |
\(0\) |
\(1\) |
\(0\) |
\(-\mathbf{x}\) |
\(\mathbf{z}\) |
\(0\) |
\(\mathbf{zx}\) |
\(\mathbf{z}\) |
\(\mathbf{z}\) |
\(0\) |
\(0\) |
\(1\) |
\(0\) |
\(-\mathbf{y}\) |
\(\mathbf{x}\) |
\(\mathbf{xy}\) |
\(\mathbf{xy}\) |
\(\mathbf{xy}\) |
\(-\mathbf{y}\) |
\(\mathbf{x}\) |
\(0\) |
\(-1\) |
\(0\) |
\(0\) |
\(-\mathbf{z}\) |
\(\mathbf{yz}\) |
\(\mathbf{yz}\) |
\(0\) |
\(-\mathbf{z}\) |
\(\mathbf{y}\) |
\(0\) |
\(-1\) |
\(0\) |
\(-\mathbf{x}\) |
\(\mathbf{zx}\) |
\(\mathbf{zx}\) |
\(\mathbf{z}\) |
\(0\) |
\(-\mathbf{x}\) |
\(0\) |
\(0\) |
\(-1\) |
\(-\mathbf{y}\) |
\(\mathbf{xyz}\) |
\(\mathbf{xyz}\) |
\(\mathbf{yz}\) |
\(\mathbf{zx}\) |
\(\mathbf{xy}\) |
\(-\mathbf{z}\) |
\(-\mathbf{x}\) |
\(-\mathbf{y}\) |
\(-1\) |