![]() ![]() Using the 2D cross product where you get (U.x*V.y-U.y*V.x) is just like doing a 3D one where the Z components are 0. the crossproduct returns a vector ortogonal to all n-1 input vectors. Note that this is also implemented in Mathematica this way. For 3 dimensions, this makes it look nice since you have only two operands. In general, for an n-dimensional cross product, you need n - 1 vectors (so that the vector orthogonal to them all can be found. Yeah its kinda weird that its so badly defined. In general, there's several analogs, and no analogs is completely equivalent.Ģ: you can define whatever function you *really* need and can use, and then use it.ģ:It is not useful to make cross product routine if you don't know it's properties.Īlso, "analog" doesn't really have any mathematical sense, just some properties is somewhat similar. (note that order of operands does not matter.)ġ: 2D cross product is not defined by itself. ![]() (3 vectors is necessary to define volume, and returned result is "normal of that volume" )Īlso, in 3D we can define another nice operation that takes 3 vectors at inputs, and return scalar, and is eqivalent to first 2D cross product analog. And other analog that takes 3 4D vectors, and have 4D vector as result. (yes, second analog takes only one argument, and return orthogonal vector :)įirst analog makes some physical and geometrical sense, second analog comes from "determinant rule", for determinant of 2x2 matrix,Īnd in 4D, there is one analog that takes 2 4D vectors at input and return 6D vector as result. and there's many other useful functions possible, that is somewhat like cross product.ĬrossProductAnalog1(U,V)=(U.x*V.y-U.y*V.x) And 3D is just only special case when both things is the same. In any other number of dimensions >=2, there's at least 2 different cross-product "analogs", depending to what properties of 3D cross product we want it to mimic. In other dimensions, cross product is not well defined, but there is well-defined analogs. Okay, maybe I'm missing something here, but doens't a cross product between two vectors with the same number of components give you another vector with that number of components? ![]()
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