## Summary of Results

We will derive the multiplication table for geometric product below, also how to calculate the inverse and transform using sandwich product. We will derive the multiplication table for outer product on this page and inner product on this page.

But first, here are the multipication tables for 1D Euclidean Space:

### table for: Geometric Product

a*b | b.1 | b.n_{0} | b.n_{∞} | b.n_{0∞} | b.n_{1} | b.n_{01} | b.n_{∞1} | b.n_{0∞1} |

a.1 | 1 | n_{0} | n_{∞} | n_{0∞} | n_{1} | n_{01} | n_{∞1} | n_{0∞1} |

a.n_{0} | n_{0} | 0 | n_{0∞}*2 | 0 | n_{01} | 0 | n_{0∞1}*2 | 0 |

a.n_{∞} | n_{∞} |
4-n_{0∞}*2 | 0 | n_{∞}*2 | n_{∞1} | n_{1}*4-n_{0∞1}*2 | 0 | n_{∞1}*2 |

a.n_{0∞} | n_{0∞} | n_{0}*2 | 0 | n_{0∞}*2 | n_{0∞1} | n_{01}*2 | 0 | n_{0∞1}*2 |

a.n_{1} | n_{1} | -n_{01} | -n_{∞1} | n_{0∞1} | 1 | -n_{0} | -n_{∞} | n_{0∞} |

a.n_{01} | n_{01} | 0 | -n_{0∞1}*2 | 0 | n_{0} | 0 | -n_{0∞}*2 | 0 |

a.n_{∞1} | n_{∞1} | -n_{1}*4+n_{0∞1}*2 | 0 | n_{∞1}*2 | n_{∞} | -1*4+n_{0∞}*2 | 0 | n_{∞}*2 |

a.n_{0∞1} | n_{0∞1} | -n_{01}*2 | 0 | n_{0∞1}*2 | n_{0∞} | -n_{0}*2 | 0 | n_{0∞}*2 |

### table for: Outer Product

a*b | b.1 | b.n_{0} | b.n_{∞} | b.n_{0∞} | b.n_{1} | b.n_{01} | b.n_{∞1} | b.n_{0∞1} |

a.1 | 1 | n_{0} | n_{∞} | n_{0∞} | n_{1} | n_{01} | n_{∞1} | n_{0∞1} |

a.n_{0} | n_{0} | 0 | n_{0∞}*2 | 0 | n_{01} | 0 | n_{0∞1}*2 | 0 |

a.n_{∞} | n_{∞} | -n_{0∞}*2 | 0 | 0 | n_{∞1} | -n_{0∞1}*2 | 0 | 0 |

a.n_{0∞} | n_{0∞} | 0 | 0 | 0 | n_{0∞1} | 0 | 0 | 0 |

a.n_{1} | n_{1} | -n_{01} | -n_{∞1} | n_{0∞1} | 0 | 0 | 0 | 0 |

a.n_{01} | n_{01} | 0 | -n_{0∞1}*2 | 0 | 0 | 0 | 0 | 0 |

a.n_{∞1} | n_{∞1} | n_{0∞1}*2 | 0 | 0 | 0 | 0 | 0 | 0 |

a.n_{0∞1} | n_{0∞1} | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

### table for: Inner Product

a*b | b.1 | b.n_{0} | b.n_{∞} | b.n_{0∞} | b.n_{1} | b.n_{01} | b.n_{∞1} | b.n_{0∞1} |

a.1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

a.n_{0} | n_{0} | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

a.n_{∞} | n_{∞} | 1*4 | 0 | 0 | 0 | 0 | 0 | 0 |

a.n_{0∞} | n_{0∞} | n_{0}*2 | 0 | 0 | 0 | 0 | 0 | 0 |

a.n_{1} | n_{1} | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

a.n_{01} | n_{01} | 0 | 0 | 0 | n_{0} | 0 | 0 | 0 |

a.n_{∞1} | n_{∞1} | -n_{1}*4 | 0 | 0 | n_{∞} | -1*4 | 0 | 0 |

a.n_{0∞1} | n_{0∞1} | -n_{01}*2 | 0 | 0 | n_{0∞} | -n_{0}*2 | 0 | 0 |

## Multiplication

Here we will derive a general expression for multiplication using the following table:

a*b |
b.n | b.n_{0} |
b.n_{∞} |
b.n_{0∞} |
b.n_{1} |
b.n_{01} |
b.n_{∞1} |
b.n_{0∞1} |

a.n | 1 | n_{0} |
n_{∞} |
n_{0∞} |
n_{1} |
n_{01} |
n_{∞1} |
n_{0∞1} |

a.n_{0} |
n_{0} |
0 | n_{0∞} |
0 | n_{01} |
0 | n_{0∞1} |
0 |

a.n_{∞} |
n_{∞} |
1-n_{0∞} |
0 | n_{∞} |
n_{∞1} |
n_{1}-n_{0∞1} |
0 | n_{∞1} |

a.n_{0∞} |
n_{0∞} |
n_{0} |
0 | n_{0∞} |
n_{0∞1} |
n_{01} |
0 | n_{0∞1} |

a.n_{1} |
n_{1} |
-n_{01} |
-n_{∞1} |
n_{0∞1} |
1 | -n_{0} |
-n_{∞} |
n_{0∞} |

a.n_{01} |
n_{01} |
0 | -n_{0∞1} |
0 | n_{0} |
0 | -n_{0∞} |
0 |

a.n_{∞1} |
n_{∞1} |
n_{0∞1}-n_{1} |
0 | n_{∞1} |
n_{∞} |
n_{0∞}-1 |
0 | n_{∞} |

a.n_{0∞1} |
n_{0∞1} |
-n_{01} |
0 | n_{0∞1} |
n_{0∞} |
-n_{0} |
0 | n_{0∞} |

Our multiplcaitiom is represented by:

**c**=**a*****b**

Where **a**, **b** and **c** are multivectors which may contain scalar, vector and bivector and trivector components.