A) \[8\text{ }sin\text{ }18{}^\circ \]
B) \[8\text{ }sin\text{ 36}{}^\circ \]
C) \[8\text{ cos 18}{}^\circ \]
D) \[8\text{ cos 36}{}^\circ \]
Correct Answer: A
Solution :
[a] \[\left| \hat{a}+\hat{b}+\hat{a}\times \hat{b} \right|\,\,=\,\,\left| \hat{a}-\hat{b} \right|\] \[\Rightarrow \,\,\,\,{{\left( \hat{a}+\hat{b} \right)}^{2}}+{{\left( \hat{a}\times \hat{b} \right)}^{2}}+2\left( \hat{a}+\hat{b} \right).\left( \hat{a}\times \hat{b} \right)\] \[={{\hat{a}}^{2}}+{{\hat{b}}^{2}}-2\hat{a}.\hat{b}\] \[\Rightarrow \,\,\,1+1+2\cos \theta +{{\sin }^{2}}\theta +0=1+1-2\cos \theta \] \[\Rightarrow \,\,\,4\cos \theta =-{{\sin }^{2}}\theta ={{\cos }^{2}}\theta -1\] \[\therefore \,\,\,\cos \theta =2-\sqrt{5}\] Now, \[{{\left| \hat{a}-\hat{b} \right|}^{2}}={{\hat{a}}^{2}}+{{\hat{b}}^{2}}-2\hat{a}.\hat{b}\] \[=1+1-2\times (2-\sqrt{5})\] \[=2(\sqrt{5}-1)=\frac{8\left( \sqrt{5}-1 \right)}{4}=8\sin 18{}^\circ \]
You need to login to perform this action.
You will be redirected in
3 sec
