Mathematics in Physics

JEE Main level, single correct, medium

10 of 121

Question 10 of 121

If two vectors A⃗\vec{A} and B⃗\vec{B} having equal magnitude RR are inclined at an angle θ\theta, then

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Answer: (C) ∣A⃗+B⃗∣=2Rcos⁡(θ2)\left|\vec{A}+\vec{B}\right|=2R\cos \left(\dfrac{\theta }{2}\right)

The magnitude of resultant vector A⃗+B⃗\vec{A}+\vec{B} is given by,
R′=A2+B2+2ABcos⁡θR'=\sqrt{{A}^{2}+{B}^{2}+2AB\cos \theta }
Here A=B=RA=B=R
Then R′=R2+R2+2R2cos⁡θR'=\sqrt{{R}^{2}+{R}^{2}+2{R}^{2}\cos \theta }
=R21+cos⁡θ=R\sqrt{2}\sqrt{1+\cos \theta }
=2R2cos⁡2θ2=\sqrt{2}R\sqrt{2{\cos }^{2}\dfrac{\theta }{2}}
=2Rcos⁡θ2=2R\cos \dfrac{\theta }{2}

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