The Orthogonal complement (or dual) of a k-blade is a (n-k)-blade where n is the number of dimensions. As the name suggests the orthogonal complement is entirely orthogonal to the corresponding k-blade. The orthogonal complement of is denoted .
In geometric algebra the orthogonal complement is found by multiplying by I which is the geometric algebra equivalent of i. In three dimensions I is a unit trivector. In two Dimensions I is a unit bivector.
In two dimensional space
- (A and B are orthogonal unit vectors)
| k-blade | Orthogonal complement | |||||
|---|---|---|---|---|---|---|
| 0-blade | Scalar | 1 | A∧B | Bivector | 2-blade | |
| 1-blade | Vector | A | B | Vector | 1-blade | |
| 2-blade | Bivector | A∧B | 1 | Scalar | 0-blade | |
In three dimensional space
- (A, B, and C are orthogonal unit vectors)
| k-blade | Orthogonal complement | |||||
|---|---|---|---|---|---|---|
| 0-blade | Scalar | 1 | A∧B∧C | Trivector | 3-blade | |
| 1-blade | Vector | A | B∧C | Bivector | 2-blade | |
| 2-blade | Bivector | A∧B | C | Vector | 1-blade | |
| 3-blade | Trivector | A∧B∧C | 1 | Scalar | 0-blade | |