If x is a unit vector such that x ( i - 2 j + k ) = - i + k , the… - Vidyadip

MCQ

If $\vec x$ is a unit vector such that $\vec x \times \left( {\hat i - 2\hat j + \hat k} \right) = - \hat i + \hat k$ , then $\vec x$ is

A
$ - \hat i$
✓
$ - \frac{1}{3}\left( {2\hat i - \hat j + 2\hat k} \right)$
C
$ \frac{1}{{\sqrt 3 }}\left( {\hat i - \hat j + \hat k} \right)$
D
$\frac{1}{3}\left( {2\hat i + \hat j + 2\hat k} \right)$

✓

Answer

Correct option: B.

$ - \frac{1}{3}\left( {2\hat i - \hat j + 2\hat k} \right)$

b
Let $\overrightarrow x = a\widehat i + b\widehat j + c\hat k$

$\left|\begin{array}{ccc}{\hat{i}} & {\hat{j}} & {\hat{k}} \\ {a} & {b} & {c} \\ {1} & {-2} & {1}\end{array}\right|=(b+2 c) \hat{i}-\hat{j}(a-c)+\hat{k}(-2 a-b)$

$ = - \widehat i + \widehat k$

$b+2 c=-1, a-c=0$ and $2 a+b=-1$

and $a^{2}+b^{2}+c^{2}=1$

$\Rightarrow a=0, b=-1, c=0$

or $\quad a=-\frac{2}{3}, b=\frac{1}{3}, c=\frac{-2}{3}$

$\Rightarrow \overline{\mathrm{x}}=-\hat{\mathrm{j}}$ or $\quad \overrightarrow{\mathrm{x}}=-\frac{1}{3}(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})$

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