Relationship between the components of tangent, normal and binormal vector.

Question

For a given arc-length parameterized curve $X(s)\in\mathbb{R}^3$ with $X_s=T\equiv(T_1, T_2, T_3)$, $N\equiv(N_1, N_2, N_3)$, $B\equiv(B_1, B_2, B_3)$, the unit tangent, normal and binormal vectors, respectively, how to obtain the relationship $T_j^2+N_j^2+B_j^2=1$, for $j=1,2,3$?

Answer 1

Let $A$ be the matrix whose columns are $T,N,B$. Then $A^\top A=I$. Since $A$ is square, it follows from basic linear algebra that $AA^\top = I$ as well. This gives you the desired equalities.