If a = i + 2 j + 3 k , b = 2 i + 3 j + k , c = 3 i + j + 2 k and … - Vidyadip

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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