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# The vector $\overrightarrow a=\alpha\hat i+2\hat j+\beta\hat k$ lies in the plane of the vectors $\overrightarrow b=\hat i+\hat j$ and $\overrightarrow c=\hat j+\hat k$ and bisects the angle between $\overrightarrow b\:\:and\:\:\overrightarrow c$, then $(\alpha,\beta)=?$

$(a)\:\:(1,1)\:\:\:\:\qquad\:\:(b)\:\:(2,2)\:\:\:\:\qquad\:\:(c)\:\:(1,2)\:\:\:\:\qquad\:\:(d)\:\:(2,1).$

Given: $\overrightarrow a=\alpha \hat i+2\hat j+\beta \hat k,\:\overrightarrow b=\hat i+\hat j\:\:\overrightarrow c=\hat j+\hat k$ are coplanar.
$\Rightarrow\:[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]=0$
$\Rightarrow\:\left |\begin {array}{ccc}\alpha & 2 & \beta\\ 1 & 1 & 0\\ 0 & 1 & 1\end {array}\right |=0$
$\Rightarrow\:\alpha-2+\beta=0$ or $\alpha+\beta=2$......(i)
Also it is given that $\overrightarrow a$ bisects the angle between $\overrightarrow b\:\:and\:\:\overrightarrow c$
$\Rightarrow\:\large\frac{\alpha+2}{|\overrightarrow a||\overrightarrow b|}=\frac{2+\beta}{|\overrightarrow a||\overrightarrow c|}$
$\Rightarrow\:\alpha=\beta$.....(ii)
From (i) and (ii) $(\alpha,\beta)=(1,1)$