A vector has a constant magnitude but its direction varies with time. What is the derivative of such a vector ?

Perpendicular to itself

Let $\vec{A}$ is a vector whose magnitude is constant but direction is changing.

Then $ \frac{d(\vec{A}.\vec{A})}{dt} = \vec{A}.\frac{d\vec{A}}{dt} + \frac{d\vec{A}}{dt}.\vec{A} = 2\vec{A}.\frac{d\vec{A}}{dt}$

$ \frac{d(\vec{A}.\vec{A})}{dt} = 0$

$\Rightarrow 2\vec{A}.\frac{d\vec{A}}{dt} = 0$

$\Rightarrow \vec{A} \text{ is perpendicular to } \frac{d\vec{A}}{dt}$