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
- Kk 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+1Ka)
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+1Ka) |
| 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 |
