3 ms·
The second term is not a cross product. It's the exterior product (or bivector). Cross product only works in 3D. Exterior product can work in any higher dimensi
by ww520 3y ago
The second term is not a cross product. It's the exterior product (or bivector). Cross product only works in 3D. Exterior product can work in any higher dimension.
A cross product of two 3D vectors is another vector, perpendicular to the plane containing the two vectors. The exterior product is a 2-vector (thus bivector) that sweeps the parallelogram between the two vectors; it's on the plane containing the two vectors. In 3D, the cross product vector is perpendicular to bivector plane.
- lupire 3y agoThe exterior product is the "correct" version of the cross product. I'm OK with the good guys seizing the name. Cross product works in dimension 2^k-1 for k in 0,1,2,3. (real / singleton, complex / duonion, quaternion, octonion) It is trivial in 0D and 1D, defined uniquely in 3D, and unique up to some arbitrary negations in 7D. https://en.m.wikipedia.org/wiki/Seven-dimensional_cross_product https://en.m.wikipedia.org/wiki/Seven-dimensional_cross_prod...
- mjcohen 3y agoNo. In n dimensions a cross product of n-1 n-vectors is the determinant with the top row being the basis elements (e_1, e_2, ..., e_n) and the next n-1 rows being the n-1 vectors. The result is a n-vector orthogonal to the n-1 vectors. In 2 dimensions it is i j x y In 3 dimensions it is i j k x1 y1 z1 x2 y2 z2 And so on.
- itishappy 3y agoThat's not a particularly fruitful generalization. The traditional definition begins: > The cross product on a Euclidean space V is a bilinear map from V × V to V. This is required for some of the cross product's useful properties such as: |a x b| = |a| |b| sin(theta) This only works in 0, 1, 3, and 7 dimensions.