A) \[\mathbf{a}+\mathbf{c}=k\,\mathbf{b}\]
B) \[\mathbf{a}+\mathbf{b}=k\,\mathbf{c}\]
C) \[\mathbf{b}+\mathbf{c}=k\,\mathbf{a}\]
D) None of these
Correct Answer: A
Solution :
Since \[\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{c}\ne \mathbf{0}\Rightarrow \mathbf{a}\times \mathbf{b}-\mathbf{b}\times \mathbf{c}=\mathbf{0}\] \[\Rightarrow \mathbf{a}\times \mathbf{b}+\mathbf{c}\times \mathbf{b}=\mathbf{0}\Rightarrow (\mathbf{a}+\mathbf{c})\times \mathbf{b}=\mathbf{0}\] \[\Rightarrow \mathbf{a}+\mathbf{c}\] is parallel to \[\mathbf{b}\Rightarrow \mathbf{a}+\mathbf{c}=k\mathbf{b}.\]
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