A) \[2\sqrt{3}\]
B) \[3\sqrt{2}\]
C) \[12\]
D) \[18\]
Correct Answer: A
Solution :
[a] \[\frac{(\vec{b}+\vec{c}).\vec{a}}{|\vec{a}|}=2|\vec{a}|\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\vec{b}+\vec{c}).\vec{a}=2{{a}^{2}}\] ??(1) \[(\vec{c}+\vec{a}).\,\,\vec{b}\,=3{{b}^{2}}\] ??(2) \[(\vec{a}+\vec{b}).\,\,\vec{c}\,=4{{c}^{2}}\] ??(3) Adding these equations, we get\ \[2(\vec{a}.\vec{b}+\vec{b}.\vec{c}+\vec{c}.\vec{a})=2{{a}^{2}}+3{{b}^{2}}+4{{c}^{2}}\] Now, \[|\vec{a}+\vec{b}+\vec{c}{{|}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2\vec{a}.\vec{b}+2\vec{b}.\vec{c}+2\vec{c}.\vec{a}\] \[=3{{a}^{2}}+4{{b}^{2}}+5{{c}^{2}}\] \[\Rightarrow \,\,\,\,|\vec{a}+\vec{b}+\vec{c}|=\sqrt{3{{a}^{2}}+4{{b}^{2}}+5{{c}^{2}}}\] \[\Rightarrow \,\,\,\,|\vec{a}+\vec{b}+\vec{c}{{|}_{\min .}}=\sqrt{3{{a}^{2}}+4{{b}^{2}}+5{{c}^{2}}}=\sqrt{12}=2\sqrt{3}\]
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