If $|\vec{A} \times \vec{B}|=\sqrt{3} \vec{A} \cdot \vec{B}$ then the value of $|\vec{A} + \vec{B}|$ is:

If $|\vec{A} \times \vec{B}|=\sqrt{3} \vec{A} \cdot \vec{B}$ then the value of $|\vec{A} + \vec{B}|$ is:
  1. $\left(A^2+B^2+A B\right)^{1 / 2}$
  2. $\left(A^2+B^2+\frac{A B}{\sqrt{3}}\right)^{1 / 2}$
  3. $A+B$
  4. $\left(A^2+B^2+\sqrt{3} A B\right)^{1 / 2}$

Solution

According to the question $\begin{aligned} & \vec{A} \times \vec{B}=\sqrt{3} \vec{A} \cdot \vec{B} \\ & A B \sin \theta =\sqrt{3} A B \cos \theta \\ & \Rightarrow \tan \theta=\sqrt{3} \\ & \Rightarrow \theta=60^{\circ} \\ & \Rightarrow|\vec{A}+\vec{B}|=\sqrt{|\vec{A}|^2+|\vec{B}|^2+2|A| |B| \cos \theta} \end{aligned}$

Asked in: NEET 2004

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