# Proof that the Frenet frame is orthonormal.

I need to prove that the vectors of the Frenet frame in $\mathbb{R}^{3}$ are orthonormal. We assume that $\alpha: [a,b]\to\mathbb{R}^{3}$ is a regular curve of unit speed, and we define the Frenet frame as $\left\lbrace \vec{T}, \vec{N}, \vec{B}\right\rbrace$, where the vectors are defined in the following manner.

$\vec{T}=\frac{\alpha'(t)}{||\alpha'(t)||},\:\:\vec{N}(t)=\frac{\vec{T}'(t)}{||\vec{T}'(t)||},\:\:\vec{B}=\vec{T}\times\vec{N}$

I know that $\vec{T}\cdot\vec{N}=0$ because the normal vector is the derivative of the tangent vector. But then how would I show that the binormal vector is orthogonal to either the tangent or the normal vector? Thanks in advance for any help!

• Seems like you need to revise the properties of the cross product. – Anthony Carapetis Sep 30 '16 at 17:48
• Yes, $\vec B$ is easy. Say again why $\vec T\cdot \vec N = 0$? Are you saying that for any particle moving, its velocity and acceleration are always orthogonal? – Ted Shifrin Oct 1 '16 at 1:18