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Which of the following relations is true for two unit vector \hat{\mathrm{A}} and \hat{\mathrm{B}} making an angle \theta to each other ?

Option: 1

|\hat{\mathrm{A}}+\hat{\mathrm{B}}|=|\hat{\mathrm{A}}-\hat{\mathrm{B}}| \tan \theta / 2


Option: 2

\begin{aligned} &|\hat{\mathrm{A}}-\hat{\mathrm{B}}|=|\hat{\mathrm{A}}+\hat{\mathrm{B}}| \tan \theta / 2 \\ \end{aligned}


Option: 3

|\hat{\mathrm{A}}+\hat{\mathrm{B}}|=|\hat{\mathrm{A}}-\hat{\mathrm{B}}| \cos \theta / 2 \\


Option: 4

|\hat{\mathrm{A}}-\hat{\mathrm{B}}|=|\hat{\mathrm{A}}+\hat{\mathrm{B}}| \cos \theta / 2


Answers (1)

best_answer


\mathrm{\left | \hat{A} +\hat{B}\right |^{2}= \left ( \hat{A} \right )^{2}+ \left ( \hat{B} \right )^{2}+2\hat{A} \: \hat{B}\: \cos \theta }
                    \mathrm{= 2+2\cos \theta }
                     \mathrm{= 2\left ( 1+\cos \theta \right ) }

\mathrm{\left | \hat{A} +\hat{B}\right |^{2}= 4\cos ^{2} \left ( \frac{\theta }{2} \right )}
\mathrm{\therefore \left | \hat{A} +\hat{B}\right |= 2\cos \left ( \frac{\theta }{2} \right )\: \rightarrow (1)}

\mathrm{\left | \hat{A} -\hat{B}\right |^{2}= \left ( \hat{A} \right )^{2}+ \left ( \hat{B} \right )^{2}+2\hat{A} \: \hat{B}\: \cos \left ( 180-\hat{\theta } \right ) }
                    \mathrm{= 1+1-2\cos \theta }
                    \mathrm{= 2\left ( 1-\cos \theta \right )}
                    \mathrm{= 4\sin ^{2}\left ( \frac{\theta }{2} \right )}

\mathrm{\left | \hat{A}-\hat{B} \right |= 2\sin \left ( \frac{\theta }{2} \right )\; \rightarrow (2)}  

\mathrm{(1)\div (2)}

\mathrm{\frac{\left | \hat{A} +\hat{B}\right |}{\left | \hat{A} -\hat{B} \right |}= \cot \left ( \frac{\theta }{2}\right)= \frac{1}{\tan \left ( \frac{\theta }{2} \right )} }
\mathrm{\left | \hat{A} -\hat{B} \right |= \tan \left ( \frac{\theta }{2} \right )\left | \hat{A} +\hat{B} \right |}
The correct option is (2)




                

Posted by

HARSH KANKARIA

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