Show that $\Vert{c(t)}\Vert$ is constant

Question

Let $c$: $I$ $\to$ $\mathbb{R}^{n}$ be a regular parametrized curve. Show that $\Vert{c(t)}\Vert$ is constant iff $c(t)$ is orthogonal to $c^{'}(t)$ for all $t$ $\in$ $I$ (Note $I$ $\subseteq$ $\mathbb{R}$).

How do I start with this problem? What is the relationship between orthogonality and the norm ($\Vert{c(t)}\Vert$) when it comes to curves?

Answer 1

Note that $||c(t)||^2=c(t)\cdot c(t)$, hence $$ \frac{d}{dt}||c(t)||^2=2c(t)\cdot c^{\prime}(t) $$ $||c(t)||$ is constant if and only if $\frac{d}{dt}||c(t)||^2=0$, and by the above equality this is true if and only if $c(t)\cdot c^{\prime}(t)=0$.