Tangent, Normal and Binormal Vectors — Question 8

Question 8

Assume $\mathbf T$ is a differentiable unit tangent and $\mathbf T'\ne\vec 0$. Prove that the Frenet vectors $\mathbf T,\mathbf N,\mathbf B$ form a right-handed orthonormal basis.

Question 8 – Solution

Strategy Differentiate $\mathbf T\cdot\mathbf T=1$, normalize $\mathbf T'$, and use cross-product identities.

Orthogonality Differentiation gives $2\mathbf T\cdot\mathbf T'=0$. Since $\mathbf N=\mathbf T'/\|\mathbf T'\|$, $\mathbf T\perp\mathbf N$, and both are unit.

Binormal Define $\mathbf B=\mathbf T\times\mathbf N$. The cross product of perpendicular unit vectors is unit and perpendicular to both.

Orientation By definition $\mathbf T\times\mathbf N=\mathbf B$, so the ordered frame is right-handed. Hence its Gram matrix is the identity and its orientation determinant is $+1$.