Question
Download Solution PDFThe value of \(\left( {\overrightarrow a - \overrightarrow b } \right).\left[ {\left( {\overrightarrow b - \overrightarrow c } \right) \times \left( {\overrightarrow c - \overrightarrow a } \right)} \right]\) is:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDF\(\left( {\vec a - \vec b} \right) \cdot \left[ {\left( {\vec b - \vec c} \right) \times \left( {\vec c - \vec a} \right)} \right]\)
\(= \left( {\vec a - \vec b} \right)\left[ {\vec b \times \vec c - \vec b \times \vec a - \vec c \times \vec c + \vec c \times \vec a} \right]\)
\(= \left( {\vec a - \vec b} \right)\left[ {\vec b \times \vec c - \vec b \times \vec a - 0 - \vec a \times \vec c} \right]\)
\(= \vec a \cdot \vec b \times \vec c - \vec a \cdot \vec b \times \vec a - \vec a \cdot \vec a \times \vec c - \vec b \cdot \vec b \times \vec c + \vec b \cdot \vec b \times \vec a + \vec b \cdot \vec a \times \vec c\)
\(= \left[ {\vec a\vec b\vec c} \right] - 0 - 0 - 0 + 0 + \left[ {\vec b\vec a\vec c} \right]\)
\(= \left[ {\vec a\vec b\vec c} \right] - \left[ {\vec a\vec b\vec c} \right]\)
Last updated on Jul 14, 2025
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