How to prove this orthogonality?

Question

In $\mathbb{R} ^3$, $v$ is a fixed vector and $\alpha:I\rightarrow \mathbb{R}$ is a regular curve. We have: $\alpha'(t)$ is orthogonal to $v$ for all $t\in I$ and $\alpha (t_0)$ is orthogonal to $v$. How can I show that $\alpha(t)$ is orthogonal to $v$ for all $t\in I$?

Answer 1

Firstly, note that two non-zero vectors are orthogonal if and only if their dot product is zero. Begin by differentiating the dot product:

$$\begin{align*} \frac{d}{dt}\alpha(t)\cdot v & = \alpha'(t)\cdot v = 0 \end{align*}$$

This means that the dot product of $\alpha$ and $v$ is constant. Since you know it is $0$ at $t_0$, it is $0$ everywhere. Thus $\alpha(t)$ is orthogonal to $v$ for all $t\in I$.