There are different difinitions of the inner product as follows:

• | Dot product | |

•_{H} |
Hestines inner product. | Like the dot product except that it is zero whenever one of its arguments is a scalar. |

left contraction inner product | ||

right contraction inner product |

These all give the same result when the operands are vectors, which means we could substitute any of them in the usual equations:

a•b = ½ (ab + ba) | This is symmetrical (a•b = b•a) |

a^b = ½ (ab - ba) | This is anti-symmetrical (a^b = - b^a) |

a * b = a•b + a^b |

Where:

- a and b are vectors
- K
_{k}a multivector of grade k

We can extend this to the multipication of a vector by a general multivector as follows:

a•K = ½ (aK + (-1)^{k+1}Ka)

a^K = ½ (aK + (-1)^{k }Ka)

a*K = a•K + a^K

Where k is the grade of K. The (-1)^{k} factor alternates the sign
as follows:

grade k | (-1)^{k} |
(-1)^{k+1} |
a•K = ½ (aK + (-1)^{k+1}Ka) |

0 (scalar) | 1 | -1 | a•K = ½ (aK - ^{}Ka) = 0 |

1 (vector) | -1 | 1 | a•K = ½ (aK + Ka) = aK |

2 (bivector) | 1 | -1 | a•K = ½ (aK - ^{}Ka) = 0 |

3 (trivector) | -1 | 1 | a•K = ½ (aK + Ka) = aK |