Tangent, Normal and Binormal Vectors — Question 4

Question 4 – Solution

Strategy. Differentiate the invariant $\mathbf T\cdot\mathbf T=1$, then use standard cross-product identities.

Step 1: Tangent–normal orthogonality Since $\mathbf T$ is a unit vector, $$\mathbf T(t)\cdot\mathbf T(t)=1.$$ Differentiate using the dot-product rule: $$\mathbf T'\cdot\mathbf T+\mathbf T\cdot\mathbf T'=0 \quad\Longrightarrow\quad 2\mathbf T\cdot\mathbf T'=0.$$ Because $\mathbf N=\mathbf T'/\|\mathbf T'\|$ and the denominator is nonzero, $$\boxed{\mathbf T\cdot\mathbf N=0}.$$

Step 2: Length of $\mathbf B$ For vectors separated by angle $\theta$, $$\|\mathbf T\times\mathbf N\|=\|\mathbf T\|\,\|\mathbf N\|\sin\theta.$$ Here both lengths are $1$ and $\theta=\pi/2$, so $$\boxed{\|\mathbf B\|=1}.$$ The cross product is perpendicular to both factors; hence $\mathbf B$ is orthogonal to both $\mathbf T$ and $\mathbf N$.

Step 3: Orientation The ordered relation $\mathbf T\times\mathbf N=\mathbf B$ is exactly the right-hand rule. Thus the three mutually perpendicular unit vectors form a right-handed orthonormal frame.