A planar curve with a nonconstant binormal vector?

Question

Does there exist a planar curve that has a non-constant binormal vector?
I thought this could not be, since the binormal vector should be perpendicular to the plane and therefore should be of the same length all the time, because the plane doesn't curve. But I was told that there were curves with a nonconstant binormal vector.
What would be an example?

Answer 1

If your curve has an inflection point, binormal vector $\vec{B}=\vec{k}$ (unit vector on the $z$ axis) switches to $-\vec{k}$ (or the inverse). In this sense, the binormal vector is not constant.

Explanation : just after the inflection point, normal vector $\vec{N}$ is the opposite of what it was just before it, therefore, $\vec{T} \times \vec{N}$ switches to $\vec{T} \times -\vec{N}$.