[Nevermark]

The normal vector is the cross product of the two vectors.

u = [u1 u2 u3] v = [v1 v2 v3]

in dimensions i, j, k

u x v = determinate of this matrix:

  =  | i  j  k|
     |u1 u2 u3|
     |v1 v2 v3|

  = (u2v3-v2y3)i - (u1v3 - v1y3)j + (u1v2 - v2u2)k

[adwn]

That works for 3D space. What about n-dimensional vectors?

[Nevermark]

You just do the cross product for n dimensions which is the determinant with n dimensions. [i j k m ...], with vector u, v, w, s, ....

Its the same for all dimensions.