Question

Q. Prove that α′(s) and α′′(s) are orthogonal.

Answer 1

Answer on Question #70541, Math / Geometry

Theorem. Prove that $\alpha'(s)$ and $\alpha''(s)$ are orthogonal.

Proof. Let $\alpha(s)$ be parametrized by arc length. It is well known that $|\alpha'(s)| = 1$.

Then $\alpha'(s) \cdot \alpha'(s) = |\alpha'(s)|^2 = 1^2 = 1$. So,

$$\left(\alpha'(s) \cdot \alpha'(s)\right)' = 1',$$ $$\alpha''(s) \cdot \alpha'(s) + \alpha'(s) \cdot \alpha''(s) = 0,$$ $$2\alpha''(s) \cdot \alpha'(s) = 0,$$ $$\alpha''(s) \cdot \alpha'(s) = 0.$$

Since $\alpha''(s) \cdot \alpha'(s) = 0$, $\alpha'(s)$ and $\alpha''(s)$ are orthogonal.

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