MHT CET · Maths · Vector Algebra
\(\overrightarrow{\mathbf{a}} \times[\overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})]\) is equal to
- A \((\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{a}}) \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})\)
- B \(\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})-\overrightarrow{\mathbf{b}}(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})\)
- C \([\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})] \overrightarrow{\mathbf{a}}\)
- D \((\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}})(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})\)
Answer & Solution
Correct Answer
(D) \((\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}})(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})\)
Step-by-step Solution
Detailed explanation
\(\overrightarrow{\mathbf{a}} \times[\overrightarrow{\mathbf{a}}\times(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})] \)
\( =\overrightarrow{\mathbf{a}} \times\{(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}) \overrightarrow{\mathbf{a}}-(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}}) \overrightarrow{\mathbf{b}}\} \)
\( =(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}})(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{a}})-(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}})(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}) \)
\( =(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}) 0+(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}})(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}}) \)
\( =(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}})(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})\)
\( =\overrightarrow{\mathbf{a}} \times\{(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}) \overrightarrow{\mathbf{a}}-(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}}) \overrightarrow{\mathbf{b}}\} \)
\( =(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}})(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{a}})-(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}})(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}) \)
\( =(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}) 0+(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}})(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}}) \)
\( =(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}})(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})\)
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