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Question: The resultant of two vectors A and B is perpendicular to the vector A and its magnitude is equal to ...

The resultant of two vectors A and B is perpendicular to the vector A and its magnitude is equal to half the magnitude of vector B. The angle between A and B is

A

120°

B

150°

C

135°

D

None of these

Answer

150°

Explanation

Solution

B2=A2+B2+2AB 6mucos⁡θ\frac{B}{2} = \sqrt{A^{2} + B^{2} + 2AB\mspace{6mu}\cos\theta} …(i)

∴ tan⁡90∘=Bsin⁡θA+Bcos⁡θ⇒A+Bcos⁡θ=0\tan 90{^\circ} = \frac{B\sin\theta}{A + B\cos\theta} \Rightarrow A + B\cos\theta = 0

∴ cos⁡θ=−AB\cos\theta = - \frac{A}{B}

Hence, from (i) B24=A2+B2−2A2⇒A=3B2\frac{B^{2}}{4} = A^{2} + B^{2} - 2A^{2} \Rightarrow A = \sqrt{3}\frac{B}{2}

⇒ cos⁡θ=−AB=−32\cos\theta = - \frac{A}{B} = - \frac{\sqrt{3}}{2}∴ θ=150∘\theta = 150{^\circ}