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10. vector algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths10. vector algebra1 Mark
MCQ
The value of $\left( {\vec a + 2\vec b - \vec c} \right).\left\{ {\left( {\vec a - \vec b} \right) \times \left( {\vec a - \vec b - \vec c} \right)} \right\}$ is equal to
  • A
    $\left[ {\vec a\,\vec b\,\vec c} \right]$
  • B
    $2\left[ {\vec a\,\vec b\,\vec c} \right]$
  • ✓
    $3\left[ {\vec a\,\vec b\,\vec c} \right]$
  • D
    $4\left[ {\vec a\,\vec b\,\vec c} \right]$
✓

Answer

Correct option: C.
$3\left[ {\vec a\,\vec b\,\vec c} \right]$
c
$(\vec{a}+2 \vec{b}-\vec{c}) \cdot\{0-\vec{a} \times \vec{b}-\vec{a} \times \vec{c}-\vec{b} \times \vec{a}+0+\vec{b} \times \vec{c}\}$

$=(\vec{a}+2 \vec{b}-\vec{c}) \cdot\{\vec{b} \times \vec{c}-\vec{a} \times \vec{c}\}$

$=[\vec{a} \vec{b} \vec{c}]-0+0-2 \vec{b} \cdot(\vec{a} \times \vec{c})-0+0$

$=[\vec{a} \vec{b} \vec{c}]-2[\vec{b} \vec{a} \vec{c}]$

$=[\vec{a} \vec{b} \vec{c}]+2[\vec{a} \vec{b} \vec{c}]=3[\vec{a} \vec{b} \vec{c}]$

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