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# If $\overrightarrow a=\hat i+\hat j+\hat k,\:\overrightarrow b=\hat i\:\:and\:\:\overrightarrow c=x\hat i-\hat j+\hat k$, then the value of $x$ for which $\overrightarrow a\:\overrightarrow b\:and\:\overrightarrow c$ are coplanar is ?

$\begin{array}{1 1} 0 \\ - 1 \\ 1 \\ \text{cannot be coplanar for any x } \end{array}$

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Toolbox:
• If $\overrightarrow a\:\overrightarrow b,\:\overrightarrow c$ are coplanar then $[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]=0$
$[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]=\left |\begin {array}{ccc}1 & 1 & 1\\1 & 0 & 0\\x & -1 & 1\end {array}\right|=2$
$\Rightarrow\:[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]\neq 0$ for any real value of $x$.

answered Dec 1, 2013