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