A) \[\sqrt{51}\]
B) \[\sqrt{37}\]
C) \[\sqrt{43}\]
D) \[\sqrt{55}\]
Correct Answer: D
Solution :
\[\left| \vec{a}+\vec{b} \right|=\sqrt{3}\]angle between \[\vec{a}\]and \[\vec{b}\]is\[{{60}^{o}}.\] \[\vec{a}x\vec{b}\]is\[{{\bot }^{r}}\] to plane containing \[\vec{a}\] and \[\vec{b}\] \[\vec{c}=\vec{a}+2\vec{b}+3\left( \vec{a}\times \vec{b} \right)\] \[\vec{c}=\sqrt{{{\left| a \right|}^{2}}+4{{\left| {\vec{b}} \right|}^{2}}+2.2{{\left| {\vec{a}} \right|}^{2}}\cos {{60}^{o}}}{{\vec{n}}_{1}}+3\left| {\vec{a}} \right|\left| {\vec{b}} \right|\sin {{60}^{o}}{{\vec{n}}_{2}}\]\[+3\left| {\vec{a}} \right|\left| {\vec{b}} \right|\sin {{60}^{o}}.{{\vec{n}}_{2}}\] \[{{\vec{n}}_{1}}{{\bot }^{r}}{{\vec{n}}_{2}}\] \[{{\left| {\vec{c}} \right|}^{2}}=\left( 1+4+2 \right)+9\times \frac{3}{4}\] \[{{\left| {\vec{c}} \right|}^{2}}=7+27/4=55/4\] \[2\left| {\vec{c}} \right|=\sqrt{55}\] So, \[h=\frac{4}{\sin {{45}^{o}}}=4\sqrt{2}\]
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