KCET · Maths · Vector Algebra
\( [\vec{a}+2 \vec{b}-\vec{c}, \vec{a}-\vec{b}, \vec{a}-\vec{b}-\vec{c}]= \)
- A \( [\vec{a}, \vec{b}, \vec{c}] \)
- B \( 3[\vec{a}, \vec{b}, \vec{c}] \)
- C \( 00 \)
- D \( 2[\vec{a}, \vec{b}, \vec{c}] \)
Answer & Solution
Correct Answer
(B) \( 3[\vec{a}, \vec{b}, \vec{c}] \)
Step-by-step Solution
Detailed explanation
(B)
\( (\vec{a}+2 \vec{b}-\vec{c}) \cdot\{\vec{a}-\vec{b} \times \vec{a}-\vec{b}-\vec{c}\} \)
\( =(\vec{a}+2 \vec{b}-\vec{c}) \cdot\{-\vec{a} \times \vec{b}-\vec{a} \times \vec{c}-\vec{b} \times \vec{a}+\vec{b} \times \vec{c}\} \)
\( =[\vec{a} \vec{b} \vec{c}]-2[\vec{b} \vec{a} \vec{c}] \)
\( =[\vec{a} \vec{b} \vec{c}]+2[\vec{a} \vec{b} \vec{c}] \)
\( =3[\vec{a} \vec{b} \vec{c}] \)
\( (\vec{a}+2 \vec{b}-\vec{c}) \cdot\{\vec{a}-\vec{b} \times \vec{a}-\vec{b}-\vec{c}\} \)
\( =(\vec{a}+2 \vec{b}-\vec{c}) \cdot\{-\vec{a} \times \vec{b}-\vec{a} \times \vec{c}-\vec{b} \times \vec{a}+\vec{b} \times \vec{c}\} \)
\( =[\vec{a} \vec{b} \vec{c}]-2[\vec{b} \vec{a} \vec{c}] \)
\( =[\vec{a} \vec{b} \vec{c}]+2[\vec{a} \vec{b} \vec{c}] \)
\( =3[\vec{a} \vec{b} \vec{c}] \)
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