Does anybody know a simple and differentiable function that converts a 3D vector u = (x, y, z)
to another vector that is orthogonal to u
.
To be more precise, I am looking for three differentiable functions {f, g, h}
such that the vector u = (x, y, z)
is orthogonal to v = (f(x,y,z), g(x,y,z), h(x,y,z))
and v
is zero only if u
is zero.
The functions {f, g, h}
should be as simple as possible. I prefer them linear, but I think no such linear functions exist. Low degree polynomials are also good.
P.S. I found such functions, but they are not polynomials. For example:
f(x, y, z) = y*(exp(x) + 3) - z*(exp(x) + 2) g(x, y, z) = z*(exp(x) + 1) - x*(exp(x) + 3) h(x, y, z) = x*(exp(x) + 2) - y*(exp(x) + 1)
It's simply the cross product of (x,y,z) with (exp(x)+1, exp(x)+2, exp(x)+3). It satisfies all requirements except for being polynomials. But they are quite simple.