Binormal vector

The third vector of the TNB frame, $\mathbf{B}=\mathbf{T}\times\mathbf{N}$ , perpendicular to both $\mathbf{T}$ and $\mathbf{N}$ .

The binormal vector $\mathbf{B}=\mathbf{T}\times\mathbf{N}$ completes the TNB frame, perpendicular to both the unit tangent and the principal unit normal. The plane spanned by $\mathbf{T}$ and $\mathbf{N}$ (perpendicular to $\mathbf{B}$ ) is the osculating plane; how fast $\mathbf{B}$ tilts as the curve twists is exactly what torsion measures.

Examples

A planar curve confined to one flat plane has a constant $\mathbf{B}$ direction the whole way along, since it never twists out of that plane.