The inner product • is an extension to the idea of the dot product in conventional vector analysis, but the dot product normally combines two vectors to give a scalar quantity, a scalar could be taken as the point at the origin so the 'meet' of two vectors is the point at the origin.

Here the • operation has been extended to higher order elements so we take two bivectors and get a vector.

There are different ways that the dot product can be extended to higher order elements such as bivectors, so we have to be careful to choose the right type to give the correct 'meet' operation.

## Contraction Inner Product

A B is a blade representing the complement (within the subspace B) of the orthogonal projection of A onto B.

Here we will attempt to build up the full multiplication table by taking each blade type in turn. We already know that the dot product of two vectors is a scalar:

ab |
b.e1 | b.e2 | b.e3 |

a.e1 | e | 0 | 0 |

a.e2 | 0 | e | 0 |

a.e3 | 0 | 0 | e |

Scalars

αβ = α β

Vector and scalar

a β = 0

Scalar and vector

α b = α b

What is the general case for multipying two bivectors?

ab |
b.e12 | b.e31 | b.e23 |

a.e12 | -e | ||

a.e31 | -e | ||

a.e23 | -e |

vector times bivector ?

a (b ∧ B) = (a b) ∧ B – b ∧(a B)

ab |
b.e12 | b.e31 | b.e23 |

a.e1 | e2 | -e3 | 0 |

a.e2 | -e1 | 0 | e3 |

a.e3 | 0 | e1 | -e2 |

full table ?

ab |
b.e | b.e1 | b.e2 | b.e3 | b.e12 | b.e31 | b.e23 | b.e123 |

a.e | e | e1 | e2 | e3 | e12 | e31 | e23 | e123 |

a.e1 | e1 | e | 0 | 0 | e2 | -e3 | 0 | e23 |

a.e2 | e2 | 0 | e | 0 | -e1 | 0 | e3 | e31 |

a.e3 | e3 | 0 | 0 | e | 0 | e1 | -e2 | e12 |

a.e12 | e12 | -e2 | e1 | 0 | -e | 0 | 0 | -e3 |

a.e31 | e31 | e3 | 0 | -e1 | 0 | -e | 0 | -e2 |

a.e23 | e23 | 0 | -e3 | e2 | 0 | 0 | -e | -e1 |

a.e123 | e123 | e23 | e31 | e12 | -e3 | -e2 | -e1 | -e |