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# If $\overrightarrow a= x^2\hat i+2\hat j-2\hat k,\:\overrightarrow b=\hat i-\hat j+\hat k\;and\:\overrightarrow c=x^2\hat i+5\hat j-4\hat k$ are three vectors, then the values of $x$ for which the angle between $\overrightarrow a\:and\:\overrightarrow b$ is acute and that between $\overrightarrow b\:and\:\overrightarrow c$ is obtuse is ?

$\begin{array}{1 1} (A) x\in (-3,3) \\(B) x\in (-3,2)\cup (2,3) \\ (C) x\in (-3,2) \\ (D) x\in [-3,3]\end{array}$

Since given that the angle between $\overrightarrow a\:and\:\overrightarrow b$ is acute and that between
$\overrightarrow b\:and\:\overrightarrow c$ is obtuse,
$\overrightarrow a.\overrightarrow b >0\: and \:\overrightarrow b.\overrightarrow c<0$
$\Rightarrow\:x^2-4 >0,\:\:and\:\:x^2 <9$
$\Rightarrow\:x\in (-\infty,-2)\:\:or\:\:(2,\infty)\:\:and\:\:x\in(-3,3)$
$\Rightarrow\:x\in(-3,-2)\cup (2,3)$