If I have these three vectors: (2,1,-4,0) , (-1,-1,2,2) , (3,2,5,4) how do I find the cross product?

Question

I know the answer is (-34, 50, -6, 11) , but how do you get that solution? what is the process?

Answer 1

If you were in R3, you'd find the cross product of 2 vectors by constructing a matrix where the 1st row is (i,j,k), the 2nd row is the components of the 1st vector, the 3rd row is the components of the 2nd vector, and then you'd find the determinant. I have to assume the same process would work here, using (i,j,k,l) as the 1st row, and so on. The i component would be the determinant of [(1,-4,0),(-1,2,2),(2,5,4)] which is -34. The j component would be opposite (sign rule) of the determinant of [(2,-4,0),(-1,2,2),(3,5,4)], which is 44, not 50. The k component (hoping only the 50 was wrong) should be det[(2,1,0),(-1,-1,2),(3,2,4)] = -6, ah, right on. The l component should be -det[(2,1,-4),(-1,-1,2),(3,2,5)] = 11.

So the cross product ought to be (-34,44,-6,11).

Turns out what I've done is find the wedge product from exterior calculus, and I lucked out that the dimension was even, since I should have put the basis vector at the bottom of the matrix. Go to the link, scroll down to wedge product.