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