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Let \mathrm{\hat{a}} and \mathrm{\hat{b}}  be two unit vectors such that \mathrm{|(\hat{a}+\hat{b})+2(\hat{a} \times \hat{b})|=2} . if \mathrm{\theta \in(0, \pi)} is the angle between  \mathrm{\hat{a}} and \mathrm{\hat{b}} then among the statements: 

(\mathrm{S} 1): 2|\hat{\mathrm{a}} \times \hat{\mathrm{b}}|=|\hat{\mathrm{a}}-\hat{\mathrm{b}}|

(\mathrm{S} 2): The projection of \mathrm{\hat{a}} on \mathrm{(\hat{a}+\hat{b})} is \mathrm{\frac{1}{2}}

Option: 1

Only (S1) is true.


Option: 2

Only (S2) is true.


Option: 3

Both (S1) and (S2) are true.


Option: 4

Both (S1) and (S2) are false.


Answers (1)

best_answer

\begin{aligned} &\mathrm{ |(\hat{a}+\hat{b})+2(\hat{a} \times \hat{b})|=2}\\ &\mathrm{\{(\hat{a}+\hat{b})+2(\hat{a} \times \hat{b})\} \cdot\{(\hat{a}+\hat{b})+2(\hat{a} \times \hat{b})\}=4}\\ &\mathrm{1+1+2 \hat{a} \cdot \hat{b}+2(\hat{a}+\hat{b}) \cdot(\hat{a} \times \hat{b})+2(\hat{a} \times \hat{b}) \cdot(\hat{a}+\hat{b})+4(\hat{a} \times \hat{b})^{2}=4}\\ &\mathrm{2 \cos \theta+0+4 \sin ^{2} \theta=4-2}\\ &\mathrm{4-4 \cos ^{2} \theta+2 \cos \theta=2}\\ &\mathrm {2 \cos ^{2} \theta-\cos \theta-1=0}\\ &\mathrm{(2 \cos \theta+1)(\cos \theta-1)=0}\\ &\mathrm{\cos \theta=1, \quad \cos \theta=-\frac{1}{2}}\\ &\mathrm{\text { (rejectad) }}\\ &\mathrm{\theta=\frac{2 \pi}{3}} \end{aligned}

\mathrm{S_{1}: \quad 2|\hat{a} \times \hat{b}|=|\hat{a}-\hat{b}|}

\mathrm {LHS: 2 \sin \frac{2 \pi}{3}=\sqrt{3} ; \quad RHS: \sqrt{1+1-2\left(-\frac{1}{2}\right)}=\\ \sqrt{3} }

\mathrm {\therefore S_{1} } is true

\mathrm{S_{2}:} Projection of \mathrm{\vec{a}} on\mathrm{(\vec{a}+\vec{b})}=     \mathrm{\frac{(\hat{a}) \cdot(\hat{a}+\hat{b})}{|\hat{a}+\hat{b}|} }

\mathrm{\begin{aligned} &=\frac{1+\left(-\frac{1}{2}\right)}{\sqrt{1+1-2\left(\frac{1}{2}\right)}} \\ &=\frac{1}{2} \end{aligned}}

\therefore \mathrm{S_{2}} is true

 

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Rishi

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