Question

A vector $\overrightarrow{a}$ is turned without a change in its length through a small angle $d\theta$. Find the value of $|\Delta \overrightarrow{a}|$ and $\Delta$a.

Answer 1

$\Delta a=$ Change in a.

$l=r\theta$

$\Rightarrow \left|\Delta a\right|=\left|a\right|d\theta$

$\Delta a=a_{final}-a_{initial}$.

$=\left|a\right|cosd\theta \cos \theta \hat{i}+\left|a\right|\sin (\theta +d\theta )\hat{j}-\left|a\right|\cos \theta \hat{i}-\left|a\right|\sin \theta \hat{j}$

$=\left|a\right|\cos \theta .\hat{i}+\left|a\right|\sin \theta -cosd\theta .\hat{j}+\left|a\right|\cos \theta sind\theta \hat{j}-\left|a\right|\cos \theta \hat{j}-\left|a\right|\sin \theta \hat{j}$

$=\left|\overrightarrow{a}\right|\sin \theta .1+\left|a\right|\cos \theta d\theta -\left|a\right|\sin \theta$

$=\left|a\right|cos\theta d\theta$.