Reading a paradox: Is the binormal vector to any vector in the boundary of a surface in the surface's tangent plane?

Question

In this picture a tangent plane is along the surface. 1: The normal vector is outward because of dividing by the norm |T| correct? 2: The orthogonal vector to them is obviously in the tangent plane by this picture, but I don't have a book right now, is that correct?

Answer 1

Dividing $T$ by $|T|$ gives you a unit tangent. It has nothing to do with whether $N$ points inward or outward.

The binormal is orthogonal to the normal, and the normal is orthogonal to the tangent plane, so the binormal must lie in the tangent plane. In the same way, a vector orthogonal to the $z$-axis must lie in the $xy$-plane.