A) \[b\times (a\times c)=0\]
B) \[a(b\times c)=0\]
C) \[c\times a=a\times b\]
D) \[c\times b=b\times a\]
Correct Answer: A
Solution :
We have \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=(\mathbf{a}\times \mathbf{b})\times \mathbf{c}\] \[\Rightarrow \,\,(\mathbf{a}\,.\,\mathbf{c})\,\mathbf{b}-(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{c}=(\mathbf{a}\,.\,\mathbf{c})\,\mathbf{b}-(\mathbf{b}\,.\,\mathbf{c})\,\mathbf{a}\] \[\Rightarrow -\,(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{c}=-\,(\mathbf{b}\,.\,\mathbf{c})\,\mathbf{a}\]\[\Rightarrow \,\,(\mathbf{b}\,.\,\mathbf{c})\,\mathbf{a}-(\mathbf{b}\,.\,\mathbf{a})\,\mathbf{c}=0\] \[\Rightarrow \,\,\mathbf{b}\times (\mathbf{a}\times \mathbf{c})=0.\]
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