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This is from a few days ago, but I am curious how one would prove this. Exercise 13.2.8 reads:

Suppose that $\left|\textbf{r}(t) \right| = k$, for some constant $k$. This means that $ \textbf{r} $ describes some path on the sphere of radius $k$ with center at the origin. Show that $ \textbf{r} $ is perpendicular to $\textbf{r}'$ at every point. Hint: Use Theorem 13.2.5, part (d).

Theorem 13.2.5 part (d) states:

$$\frac{d}{dt} (\textbf{r}(t) \cdot \textbf{s}(t)) = \textbf{r}'(t) \cdot \textbf{s}(t) + \textbf{r}(t) \cdot \textbf{s}'(t) $$

I'm just not sure where to begin. If someone could give me a starting point, I would really appreciate it!

This is from a few days ago, but I am curious how one would prove this. Exercise 13.2.8 reads:

Suppose that $\left|\textbf{r}(t) \right| = k$, for some constant $k$. This means that $ \textbf{r} $ describes some path on the sphere of radius $k$ with center at the origin. Show that $ \textbf{r} $ is perpendicular to $\textbf{r}'$ at every point. Hint: Use Theorem 13.2.5, part (d).

Theorem 13.2.5 part (d) states:

$$\frac{d}{dt} (\textbf{r}(t) \cdot \textbf{s}(t)) = \textbf{r}'(t) \cdot \textbf{s}(t) + \textbf{r}(t) \cdot \textbf{s}'(t) $$

I'm just not sure where to begin. If someone could give me a starting point, I would really appreciate it!