Question
Download Solution PDFIf \(\rm \vec{a},\vec{b},\vec{c}\) are three non-coplanar vectors, then
\(\rm (\vec{a}+\vec{b}+\vec{c}) \cdot[(\vec{a}+\vec{b}) \times( \vec{a}+\vec{c})]=\)
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
If \(\rm \vec P, \vec Q, \vec R\) are vectors, then
- \(\rm \vec P \times \vec P = 0\)
- \(\rm \vec P\cdot(\rm \vec P \times \text{any vector}) =0\)
- \(\rm \vec P\cdot (\vec Q\times\vec R) = \vec Q\cdot (\vec R\times\vec P)=\vec R\cdot( \vec P\times\vec Q) =[ \vec P, \vec Q, \vec R]\)
- For dot product \(\rm (\vec P + \vec Q) \cdot \vec R = (\vec P\cdot \vec R) +(\vec Q\cdot \vec R)\)
- For cross product \(\rm (\vec P + \vec Q) \times\vec R = (\vec P\times\vec R) + (\vec Q\times\vec R) \)
Calculation:
\(\rm S=(\vec{a}+\vec{b}+\vec{c}) \cdot[(\vec{a}+\vec{b}) \times( \vec{a}+\vec{c})]\)
⇒ \(\rm S=(\vec{a}+\vec{b}+\vec{c}) \cdot[(\vec{a}\times\vec{a}+\vec{a}\times\vec{c}+\vec{b}\times\vec{a}+\vec{b}\times\vec{c})]\)
⇒ \(\rm S=(\vec{a}+\vec{b}+\vec{c}) \cdot[(0+\vec{a}\times\vec{c}+\vec{b}\times\vec{a}+\vec{b}\times\vec{c})]\)
\(\rm S=\vec{a}\cdot[\vec{a}\times\vec{c}+\vec{b}\times\vec{a}+\vec{b}\times\vec{c}]+\vec{b}\cdot[\vec{a}\times\vec{c}+\vec{b}\times\vec{a}+\vec{b}\times\vec{c}]+\vec{c}\cdot[\vec{a}\times\vec{c}+\vec{b}\times\vec{a}+\vec{b}\times\vec{c}] \)
\(\rm S=\vec{a}\cdot(\vec{a}\times\vec{c})+\vec{a}\cdot(\vec{b}\times\vec{a})+\vec{a}\cdot(\vec{b}\times\vec{c})+\vec{b}\cdot(\vec{a}\times\vec{c})+\vec{b}\cdot(\vec{b}\times\vec{a})+\vec{b}\cdot(\vec{b}\times\vec{c})+\vec{c}\cdot(\vec{a}\times\vec{c})+\vec{c}\cdot(\vec{b}\times\vec{a})+\vec{c}\cdot(\vec{b}\times\vec{c})\)
⇒ \(\rm S=0+0+\vec{a}\cdot(\vec{b}\times\vec{c})+\vec{b}\cdot(\vec{a}\times\vec{c})+0+0+0+\vec{c}\cdot(\vec{b}\times\vec{a})+0\)
⇒ \(\rm S=\vec{a}\cdot(\vec{b}\times\vec{c})+\vec{b}\cdot(\vec{a}\times\vec{c})+\vec{c}\cdot(\vec{b}\times\vec{a}) \)
⇒ \(\rm S=\vec{a}\cdot(\vec{b}\times\vec{c})+\boldsymbol{\rm [-\vec{b}\cdot(\vec{c}\times\vec{a})]+[-\vec{c}\cdot(\vec{a}\times\vec{b})]}\)
⇒ \(\rm S=[\vec{a}, \vec{b}, \vec{c}]-[\vec{a}, \vec{b}, \vec{c}]-[\vec{a}, \vec{b}, \vec{c}]\)
∴ \(\boldsymbol{\rm S=-[\vec{a}, \vec{b}, \vec{c}]} \)
Last updated on Jun 12, 2025
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