FANDOM


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 $ \mathbf{A} $ is denoted $ \overline{\mathbf{A}} $.

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
Community content is available under CC-BY-SA unless otherwise noted.