# Differential Geometry/Binormal Vector, Binormal Line, and Rectifying Plane

Consider a curve C of class of at least 2 with the arc length parametrization f(s).

The unit binormal vector is the cross product of the unit tangent vector and the unit principal normal vector,

$b(s)=t(s)\times p(s)$

which has a magnitude of 1 because t(s) and p(s) are orthogonal, and which are orthogonal to both t(s) and p(s).

The line passing through f(s) in the direction of b(s) is called the binormal line, and the plane spanned by the b(s) and t(s) is called the rectifying plane.

Now we have equations of the three planes. The normal plane is given by the equation $(x_1-f(s))\cdot t(s)=0$, the rectifying plane is given by the equation $(x_1-f(s))\cdot p(s)=0$, and the osculating plane is given by the equation $(x_1-f(s))\cdot b(s)=0$. It is easy to see that the earlier formula for the osculating plane is the same as this formula.