A) \[a=1,\,\,b=1\]
B) \[c=1,\,\,a=1\]
C) \[\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=1\]
D) \[b=1,\,\,c=a\]
Correct Answer: D
Solution :
Since,\[\mathbf{a}\times \mathbf{b}=\mathbf{c}\] \[\therefore \] \[(\mathbf{b}\times \mathbf{c})\times \mathbf{b}=\mathbf{c}\] \[\Rightarrow \] \[(\mathbf{b}\cdot \mathbf{b})\mathbf{c}-(\mathbf{b}\cdot \mathbf{c})\mathbf{b}=\mathbf{c}\] \[\Rightarrow \] \[(\mathbf{b}\cdot \mathbf{b})=1\] \[\mathbf{c}\cdot \mathbf{b}=0\] \[\Rightarrow \] \[{{b}^{2}}=1\] \[i.e.,\,\,b=1\] \[\therefore \] \[\mathbf{b}\]is a unit vector \[\therefore |\mathbf{c}|\,\,=\,\,|\mathbf{a}|\] \[\Rightarrow \] \[c=a\]
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