A) \[\pm \frac{2}{\sqrt{3}}\,(a\times b)\]
B) \[\pm \frac{\sqrt{3}}{2}\,(a\times b)\]
C) \[\pm \,2\,\,(a\times b)\]
D) \[\pm \,\,\frac{1}{2}\,(a\times b)\]
Correct Answer: A
Solution :
Given that, a, b, and c are unit vectors. \[\Rightarrow \] \[|a|=|b|=|c|=1\] and \[b\,.\,c=a\,.\,c=0\] ie., vector c is perpendicular to both vectors a and b. So, the unit vector \[c=\frac{(a\times b)}{|a\times b|}\] ?.(i) We know that, \[{{(a\times b)}^{2}}=|a{{|}^{2}}\,|b{{|}^{2}}-{{(a\,.\,b)}^{2}}\] \[={{(1)}^{2}}.{{(1)}^{2}}-{{(|a|\,|b|\,\cos \,\pi /3)}^{2}}\] (given) \[=1-{{\left( 1.\frac{1}{2} \right)}^{2}}=1-\frac{1}{4}=\frac{3}{4}\] \[\Rightarrow \] \[|(a\times b){{|}^{2}}=\frac{3}{4}\,\,\,\Rightarrow \,\,|a\times b|\,\,=\pm \,\frac{\sqrt{3}}{2}\] From Eq. (i), \[c=\pm \frac{2}{\sqrt{3}}\,\,(a\times b)\]
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