Q)

Let γ:I->R3 be a regular parameterized curve, with non zero curvature everywhere and arc length s as parameter.

Let σ(s,v) = γ(s) + r(n(s) cos v +b(s) sin v) [where r is a non zero constant] be a parameterized surface, where n is the normal vector and b is the binormal vector of γ.

Show that when σ is regular, its unit normal vector is:

N(s,v) = -(n(s) cos v +b(s) sin v)

Answer)

Well, the definition of a unit normal vector is:

N(s,v) = [∂σ/∂s X ∂σ/∂v]/[|∂σ/∂s X ∂σ/∂v|]

So then I proceed to evaluate ∂σ/∂s and ∂σ/∂v..

∂σ/∂s = ∂γ/∂s + r[(dn/ds)(cos v) + (db/ds)(sinv)]

and

∂σ/∂v = r[-n(s)sin v + b(s)cosv]

But now I'm wondering, how do I cross product these two if they are not vectors!!