k-vectors (Multivector of Grade k)
- is a linear combination of k-blades
- is a homogeneous multivector of grade k
- is dual of a differential k-form
k-vector vs multivector vs k-blade
See: k-blades vs k-vectors vs multivectors
Examples
- 0-vectors are scalars
- 1-vectors are vectors
- 2-vectors are bivectors
- 3-vectors are trivectors
They are respectively dual to 0-forms, 1-forms, 2-forms, and 3-forms
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Orientation is defined by an ordered set of vectors. |
Reversed orientation corresponds to negating the exterior product. |
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Geometric interpretation of grade k elements in a real exterior algebra for
The exterior product of k vectors can be visualized as any k-dimensional shape (e.g. k-parallelotope, k-ellipsoid); with magnitude (hypervolume), and orientation defined by that on its (k − 1)-dimensional boundary and on which side the interior is. | |
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Other
The highest grade element in a space is called a pseudoscalar.
- (n − 1)-vectors are pseudovectors
- n-vectors are pseudoscalars