It is correct that the determinant method would only work if the vectors lie in 3D space?
keenan
The determinant is well-defined in any dimension, but formulas involving the cross product will work only in 3D.
Azure
I don't think this is true; there should be more than one vector that satisfies this. Considering that u x v is the actual cross product, since the determinant is a multilinear function, and det(u,v,v) = 0, then det(u,v,uxv) = det(u,v,uxv)+0 = det(u,v,uxv) + det(u,v,v) = det(u,v,uxv+v), constructing another vector that has the same requirements.
I think that in order to truly limit it to one vector, we also have to put in the conditions that u.dot(u.cross(v)) = v.dot(u.cross(v)) = 0... which still doesn't work if v and u are parallel.
triangle
Based on this unique formula, we could also express this cross product u x v by using a matrix to represent the coefficients in u and multiplying by the vector v.
small_potato__
Fun fact: earlier versions of Matlab allowed you to take a cross product in multiple dimensions, but in recent updates, they have taken this away and made it only for vectors of length 3.
It is correct that the determinant method would only work if the vectors lie in 3D space?
The determinant is well-defined in any dimension, but formulas involving the cross product will work only in 3D.
I don't think this is true; there should be more than one vector that satisfies this. Considering that u x v is the actual cross product, since the determinant is a multilinear function, and det(u,v,v) = 0, then det(u,v,uxv) = det(u,v,uxv)+0 = det(u,v,uxv) + det(u,v,v) = det(u,v,uxv+v), constructing another vector that has the same requirements.
I think that in order to truly limit it to one vector, we also have to put in the conditions that u.dot(u.cross(v)) = v.dot(u.cross(v)) = 0... which still doesn't work if v and u are parallel.
Based on this unique formula, we could also express this cross product u x v by using a matrix to represent the coefficients in u and multiplying by the vector v.
Fun fact: earlier versions of Matlab allowed you to take a cross product in multiple dimensions, but in recent updates, they have taken this away and made it only for vectors of length 3.