A) \[8\overrightarrow{\alpha }.(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\]
B) \[\overrightarrow{\alpha }.(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\]
C) \[8(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\]
D) None of these
Correct Answer: A
Solution :
We have, \[\overrightarrow{\alpha }=x(\overrightarrow{a}\times \overrightarrow{b})+y(\overrightarrow{b}\times \overrightarrow{c})+z(\overrightarrow{c}\times \overrightarrow{a})\] Taking dot product with\[\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\]respectively, we get \[\overrightarrow{\alpha }.\overrightarrow{a}=y[\overrightarrow{a}\,\,\overrightarrow{b}\,\,\overrightarrow{c}]\Rightarrow y=8(\overrightarrow{\alpha }.\overrightarrow{a})\] \[\overrightarrow{\alpha }.\overrightarrow{b}=z((\overrightarrow{c}\times \,\overrightarrow{a}).\,\overrightarrow{b})\] \[\Rightarrow \] \[\overrightarrow{\alpha }.\overrightarrow{b}=z[\overrightarrow{a}\,\,\overrightarrow{b}\,\,\overrightarrow{c}]\Rightarrow z=8(\overrightarrow{\alpha }.\overrightarrow{b})\] and \[\overrightarrow{\alpha }.\overrightarrow{c}=a(\overrightarrow{a}\times \,\overrightarrow{b}.\,\overrightarrow{c})\] \[\overrightarrow{\alpha }.\overrightarrow{c}=x[\overrightarrow{a}\,\,\overrightarrow{b}\,\,\overrightarrow{c}]\Rightarrow x=8(\overrightarrow{\alpha }.\overrightarrow{c})\] \[\therefore \] \[x+y+z=8\overrightarrow{\alpha }.(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\]
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