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10. vector algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths10. vector algebra1 Mark
MCQ
If $\vec a = \vec i + 2\vec j + 3\vec k$ , $\vec b = 2\vec i + 3\vec j + \vec k$ , $\vec c = 3\vec i + \vec j + 2\vec k$ and $\alpha \vec a + \beta \vec b + \gamma \vec c =  - 3\left( {\hat i - \hat k} \right)$ . Then the triplet $\left( {\alpha ,\beta ,\gamma } \right)$ is
  • ✓
    $\left( {2, - 1, - 1} \right)$
  • B
    $\left( {-2, 1, 1} \right)$
  • C
    $\left( {-2, - 1, 1} \right)$
  • D
    $\left( {2, 1, - 1} \right)$
✓

Answer

Correct option: A.
$\left( {2, - 1, - 1} \right)$
a
$\alpha \overrightarrow{\mathrm{a}}+\beta \overrightarrow{\mathrm{b}}+\gamma \overrightarrow{\mathrm{c}}=-3(\hat{\mathrm{i}}-\hat{\mathrm{k}})$

$\alpha(1,2,3)+\beta(2,3,1)+\gamma(3,1,2)=-3(1,0,-1)$

$\Rightarrow \alpha+2 \beta+3 \gamma=-3$

$\Rightarrow 2 \alpha+3 \beta+\gamma=0$

$\Rightarrow 3 \alpha+\beta+2 \gamma=3$

$\Rightarrow \alpha=2, \beta=-1, \gamma=-1$

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