A) \[{{\sin }^{2}}A+{{\sin }^{2}}B\text{ }+{{\sin }^{2}}C\]
B) \[\frac{3\sqrt{3}}{2}\]
C) 2
D) 3
Correct Answer: B
Solution :
We have, \[a\times b=\left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 2 & 1 & -2 \\ 1 & 1 & 0 \\ \end{matrix} \right|=2\hat{i}-2\hat{j}+\hat{k}\]and \[|a\times b|=\sqrt{{{2}^{2}}+{{2}^{2}}+{{1}^{2}}}=\sqrt{4+4+1}=3\] \[\therefore \]\[|(a\times b)\times c|=|a\times b|\,\,|c|\sin {{30}^{o}}\] \[=\frac{3}{2}|c|\] Given, \[|c-a|=2\sqrt{2}\] \[\Rightarrow \]\[|c-a{{|}^{2}}=8\] \[\Rightarrow \]\[|c{{|}^{2}}+|a{{|}^{2}}-2(a.c)=8\] \[\Rightarrow \]\[|c{{|}^{2}}+9-2|c|=8\] \[\Rightarrow \]\[|c{{|}^{2}}-2|c|+1=0\]\[\Rightarrow \]\[|c|=1\] Hence, \[|(a\times b)\times c|=\frac{3}{2}\]You need to login to perform this action.
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