Vector Algebra Question 186
Question: $ [(\mathbf{a}\times \mathbf{b})\times (\mathbf{b}\times \mathbf{c}),(\mathbf{b}\times \mathbf{c})\times (\mathbf{c}\times \mathbf{a}),(\mathbf{c}\times \mathbf{a})\times (\mathbf{a}\times \mathbf{b})]=, $
Options:
A) $ {{[\mathbf{a}\mathbf{b}\mathbf{c}]}^{2}} $
B) $ {{[\mathbf{a}\mathbf{b}\mathbf{c}]}^{3}} $
C) $ {{[\mathbf{a}\mathbf{b}\mathbf{c}]}^{4}} $
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
- We have $ (\mathbf{a}\times \mathbf{b})\times (\mathbf{b}\times \mathbf{c}) $
$ =((\mathbf{a}\times \mathbf{b}),.,\mathbf{c})\mathbf{b}-((\mathbf{a}\times \mathbf{b}),.,\mathbf{b})\mathbf{c}=[\mathbf{a},\mathbf{b},\mathbf{c}]\mathbf{b} $
$ (\mathbf{b}\times \mathbf{c})\times (\mathbf{c}\times \mathbf{a})=((\mathbf{b}\times \mathbf{c}),.,\mathbf{a})\mathbf{c}-((\mathbf{b}\times \mathbf{c}),.,\mathbf{c})\mathbf{a}=[,\mathbf{b},\mathbf{c},\mathbf{a}],\mathbf{c} $
$ (\mathbf{c}\times \mathbf{a})\times (\mathbf{a}\times \mathbf{b})=((\mathbf{c}\times \mathbf{a}),.,\mathbf{b})\mathbf{a}-((\mathbf{c}\times \mathbf{a}),.,\mathbf{a})\mathbf{b}=[\mathbf{c},\mathbf{a},\mathbf{b}],\mathbf{a} $
$ \therefore ,[(\mathbf{a}\times \mathbf{b})\times (\mathbf{b}\times \mathbf{c})(\mathbf{b}\times \mathbf{c})\times (\mathbf{c}\times \mathbf{a})(\mathbf{c}\times \mathbf{a})\times (\mathbf{a}\times \mathbf{b})] $
$ =[[\mathbf{a},\mathbf{b},\mathbf{c}],\mathbf{a},[\mathbf{a},\mathbf{b},\mathbf{c}],\mathbf{b},[\mathbf{a},\mathbf{b},\mathbf{c}],\mathbf{c}] $ $ ={{[\mathbf{a},\mathbf{b},\mathbf{c}]}^{3}}[\mathbf{a},\mathbf{b},\mathbf{c}]={{[\mathbf{a},\mathbf{b},\mathbf{c}]}^{4}}. $
BETA
