MHT CET · Maths · Vector Algebra
For any non-zero vectors \(\bar{a}, \bar{b}, \bar{c}\), the value \(\overline{\mathrm{a}} \cdot[(\overline{\mathrm{b}} \times \overline{\mathrm{c}}) \times(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}})]\) is
- A \(2[\bar{a} \bar{b} \bar{c}]\)
- B \([\bar{a} \bar{b} \bar{c}]\)
- C \([\overline{\mathrm{a}} \overline{\mathrm{c}} \overline{\mathrm{b}}]\)
- D 0
Answer & Solution
Correct Answer
(D) 0
Step-by-step Solution
Detailed explanation
\(\bar{a} \cdot[(\bar{b} \times \bar{c}) \times(\bar{a}+\bar{b}+\bar{c})] \)
\( =\bar{a} \cdot[(\bar{b} \times \bar{a})+(\bar{b} \times \bar{b})+(\bar{b} \times \bar{c})+(\bar{c} \times \bar{a})+(\bar{c} \times \bar{b})+(\bar{c} \times \bar{c})] \)
\( =\bar{a} \cdot(\bar{b} \times \bar{a})+\bar{a} \cdot(0)+\bar{a} \cdot(\bar{b} \times \bar{c})+\bar{a} \cdot(\bar{c} \times \bar{a})+\bar{a} \cdot(\bar{c} \times \bar{b})+\bar{a} \cdot(0) \)
\( =0+\bar{a} \cdot(\bar{b} \times \bar{c})+0-\bar{a} \cdot(\bar{b} \times \bar{c})=0\)
\( =\bar{a} \cdot[(\bar{b} \times \bar{a})+(\bar{b} \times \bar{b})+(\bar{b} \times \bar{c})+(\bar{c} \times \bar{a})+(\bar{c} \times \bar{b})+(\bar{c} \times \bar{c})] \)
\( =\bar{a} \cdot(\bar{b} \times \bar{a})+\bar{a} \cdot(0)+\bar{a} \cdot(\bar{b} \times \bar{c})+\bar{a} \cdot(\bar{c} \times \bar{a})+\bar{a} \cdot(\bar{c} \times \bar{b})+\bar{a} \cdot(0) \)
\( =0+\bar{a} \cdot(\bar{b} \times \bar{c})+0-\bar{a} \cdot(\bar{b} \times \bar{c})=0\)
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