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3-1.Vectors
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If $\vec A$ and $\vec B$ are two non-zero vectors such that $\left| {\vec A + \vec B} \right| = \frac{{\left| {\vec A - \vec B} \right|}}{2}$ and $\left| {\vec A} \right| = 2\left| {\vec B} \right|$ then the angle between $\vec A$ and $\vec B$ is
A
$37^o$
B
$53^o$
C
$cos^{-1}(-3/4)$
D
$cos^{-1}(-4/3)$
Solution
$\sqrt{A^{2}+B^{2}+2 A B \cos \theta}=\frac{1}{2} \times \sqrt{A^{2}+B^{2}-A B \cos \theta}$
$4\left(A^{2}+B^{2}+2 A B \cos \theta\right)=A^{2}+B^{2}-2 A B \cos \theta$
$3 A^{2}+3 B^{2}+10 A B \cos \theta=0$
$12 \mathrm{B}^{2}+3 \mathrm{B}^{2}+20 \mathrm{B}^{2} \cos \theta=0$
$20 \mathrm{B}^{2} \cos \theta=-15 \mathrm{B}^{2}$
$\cos \theta=-\frac{3}{4}$
$\theta=\cos ^{-1}\left(-\frac{3}{4}\right)$
Standard 11
Physics
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