• # question_answer $[(\mathbf{a}\times \mathbf{b})\times (\mathbf{b}\times \mathbf{c})\,(\mathbf{b}\times \mathbf{c})\times (\mathbf{c}\times \mathbf{a})\,(\mathbf{c}\times \mathbf{a})\times (\mathbf{a}\times \mathbf{b})]=\,$ 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

• 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}}.$