A vector $\overrightarrow{a}$ is turned without a change in its length through a small angle $d\theta$. The value of $\left|\mathrm{\Delta }\overrightarrow{a}\right|$ and $∆a$ are respectively

From the figure $\left|\overrightarrow{OA}\right|=a\text{ and }\left|\overrightarrow{OB}\right|=a$

Also from triangle rule $\overrightarrow{OB}-\overrightarrow{OA}=\overrightarrow{AB}=\mathrm{\Delta }\overrightarrow{a}$

$\begin{array}{l}\Rightarrow \left|\mathrm{\Delta }\overrightarrow{a}\right|=AB\\ \text{ Using, angle }=\frac{\text{ arc }}{\text{ radius }}\\ \Rightarrow AB=a\cdot d\theta \\ \text{ So, }|∆\overrightarrow{a}|=ad\theta \end{array}$

$\begin{array}{l}\mathrm{\Delta }a\text{ means change in magnitude of vector, i.e., }\left|\overrightarrow{OB}\right|-\left|\overrightarrow{OA}\right|\\ \Rightarrow a-a=0\\ \text{ So }\mathrm{\Delta }a=0\end{array}$