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Explain under what conditions, the resultant of two vectors will be equal to either of them.

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The resultant of two vectors will be equal to either of them if :

(i) The two vectors will be equal to either of them if :

(ii) The two vectors are inclined to each other at an angle of 120º.

Explanation – Let X be magnitude of each of the two vectors say \(\vec P\) and \(\vec Q\) . if θ = angle between \(\vec P\) and \(\vec Q\) , then the magnitude of their resultant vector (\(\vec R\) ) is given by the relation

R = \(\sqrt{P^2+Q^2+2PQ\,cos\,\theta}\)

= \(\sqrt{X^2+X^2+2X^2\,cos\,120^o}\)

\(\sqrt{2X^2[1+cos(90^o+30^o)]}\)

= \(\sqrt{2X^2[1-sin\,30^o]}\)

= \(\sqrt{2X^2(1-\frac{1}{2})}\)   (∴ sin 30º = \(\frac{1}{2}\))

= \(\frac{2X^2}{2}\) = X

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