α is a plane curve iff the binormal vector is constant

Question

I've just started studing Differencial Geometry at college and I came across the following exercise

"α is a plane curve iff the binormal vector is constant"

Would you have any hints for this proof?

I know that if the binormal vector is constant then the curve is in the span generated by T(s) and N(s)

Answer 1

The result you quoted gets one direction, ie if the curve lies in the span of two vectors then it's a plane curve. For the other direction, assuming $\alpha$ is a plane curve then you can use the definition of binormal vector to show that it must be constant.