Question

If $x$ is parallel to $y$ and $z$, where $x=2\hat{i}+\hat{j}+\alpha \hat{k},y=\alpha \hat{i}+\hat{k}$ and $z=5\hat{i}-\hat{j}$, then $\alpha$ is equal to

• A. $\pm \sqrt{5}$
• B. $\pm \sqrt{6}$
• C. $\pm \sqrt{7}$
• D. None of these

Answer 1

The correct option is C $\pm \sqrt{7}$

Given, $x$ is parallel to $y$ and $z$
Therefore, $x\cdot (y\times z)=0$
$\Rightarrow \left[\begin{array}{ccc}x& y& z\end{array}\right]=0\Rightarrow \left|\begin{array}{ccc}2& 1& \alpha \\ \alpha & 0& 1\\ 5& -1& 0\end{array}\right|=0$
$\Rightarrow 2(0+1)-1(0-5)+\alpha (-\alpha -0)=0$
$\Rightarrow 2+5-{\alpha }^2=0$
$\Rightarrow {\alpha }^2=7$
$\Rightarrow \alpha =\pm \sqrt{7}$