Math Question---Cross Product of a 4 Unit Vector?!!?-?!?

[TangDN]

How do I take the cross product of two vectors? I understand how to take it of two 3 unit vectors... But what's the math to multiply two 4 unit vectors??? Thanks!

 

[silverpig]

The cross product (generally) only applies to 3-space. It constructs a vector perpendicular to the original two. If you tried to cross product two vectors in 4-space, you'd get a plane perpendicular to both vectors. I guess you could take the cross product of 3 vectors in 4 space to get a 4th vector perpendicular to the other 3, but I've never heard of that being asked before... FYI:

(notation: u1 = a, u2 = b, u3 = c; v1 = d, v2 = e, v3 = f; where the u1, v2, etc are components of vectors u and v. i, j, k, are the unit vectors in the standard directions)

u x v =

ijk
abc
def

The cross product is the determinant of this 3x3 matrix. That is you add the products of all the diagonals that go to the right, and subtract the products of the diagonals that go to the left... ibf + jcd + kae - ice - jaf - kbd = u x v

I'm gonna go out on a limb and guess that the cross product of 2 vectors in 4 space would be the determinant of

ijkl
qrst
abcd
efgh

where i, j, k, l are the unit vectors; q, r, s, t are parameters; a, b, c, d are components of vector 1; and e, f, g, h are components of vector 2...

(note, the diagonal multiplication deal for finding the determinant won't work here... You gotta use cofactor expansion, or the super long inversion method...)