Question

Write an expression for the magnitude of the resultant vector 'R' of two vectors $\overrightarrow{A}and\overrightarrow{B}$ acting at a point. When will this resultant vector 'R' be maximum?

Answer 1

Resultant vector,

$\left|\overrightarrow{R}\right|=\sqrt{{\left|\overrightarrow{A}\right|}^2+{\left|\overrightarrow{B}\right|}^2+2\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|\cos \theta }$

Where,

$\theta$ is the angle between them.

For finding maximum.

$\frac{d\left|\overrightarrow{R}\right|}{d\theta }=\frac{2\sin \theta }{2\sqrt{{\left|\overrightarrow{A}\right|}^2+{\left|\overrightarrow{B}\right|}^2+2\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|\cos \theta }}=0$

$\sin \theta =0$

$\theta =\pi n(n=0,1,\mathrm{2....})$

$\cos \left(n\pi \right)=1$ (when, $n=0,2,4....$ )

Hence, when both vector in same direction, resultant will be maximum.