JEE Main & Advanced Physics Vectors Question Bank Topic Test - Vectors

A vector \[\vec{a}\] is turned without a change in its length through a small angle \[d\theta \]. The value of \[|\Delta \vec{a}|\] and \[\Delta a\] are respectively

A) \[0,\,a\,d\theta \]

B) \[a,\,d\theta ,\,\,0\]

C) 0, 0

D) None of these

Correct Answer: D

Solution :

[d] \[|\Delta \vec{a}|=\,\sqrt{{{a}^{2}}+\,{{a}^{2}}-\,2.a.a.\cos \,(d\,\theta )}\]

\[=2a\,\,\sin \,\left( \frac{d\theta }{2} \right)\]

For small angle

\[\sin \,\frac{d\theta }{2}\,\,\approx \,\,\frac{d\theta }{2}\]

\[\therefore \] \[|\Delta \vec{a}|\,\,=2a\,\times \,\frac{d\theta }{2}=\,a.d\theta \]