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# If $\overrightarrow a=\hat i+\hat j+\hat k\:\:\overrightarrow b=4\hat i+3\hat j+4\hat k\:\:and\:\:\overrightarrow c=\hat i+\alpha\hat j+\beta\hat k$ are linear dependent vectors and $|\overrightarrow c|=\sqrt 3$ then $(\alpha,\beta)=?$

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• If $\overrightarrow a, \:\overrightarrow b\:\;and\:\:\overrightarrow c$ are linearly dependent, then $[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]=0$
Given: $\overrightarrow a, \:\overrightarrow b\:\;and\:\:\overrightarrow c$ are linearly dependent,
$\Rightarrow\:\left |\begin {array}{ccc}1 & 1 & 1\\ 4 & 3 & 4\\1 & \alpha & \beta\end {array}\right|=0$
$\Rightarrow\:3\beta-4\alpha-4\beta+4+4\alpha-3=0$
$\Rightarrow\:1-\beta=0$ or $\beta=1$
Also given that $|\overrightarrow c|=\sqrt 3$
$\Rightarrow\:1+\alpha^2+\beta^2=3$
$\Rightarrow\:\alpha=\pm 1$
$\therefore\:(\alpha,\beta)=(\pm 1,1)$