Question : Find the unit vector perpendicular to the plane of two vectors $\displaystyle \bar{a}$ and $\displaystyle \bar{b}$ ,
$\displaystyle \bar{a} = \hat{i} - \hat{j} + 2\hat{k}$ and $\displaystyle \bar{b} = 2\hat{i} + 3\hat{j} - \hat{k}$
Question : Find the unit vector perpendicular to the plane of two vectors $\displaystyle \bar{a}$ and $\displaystyle \bar{b}$ ,
$\displaystyle \bar{a} = \hat{i} - \hat{j} + 2\hat{k}$ and $\displaystyle \bar{b} = 2\hat{i} + 3\hat{j} - \hat{k}$
$\displaystyle | \bar a \times \bar b|$ = $\displaystyle \begin{vmatrix}i & j & k \\ 1 & -1 & 2 \\ 2 & 3 & -1 \end{vmatrix}$ = $\displaystyle -5i + 5j +5k$
$\displaystyle | \bar a \times \bar b|$ = $\displaystyle \sqrt{ (- 5)^2 + 5^2 + 5^2 }$ = $\displaystyle \sqrt{75}$ = $\displaystyle 5 \sqrt{3}$
$\displaystyle u_n$ = $\displaystyle \frac{\bar a \times \bar b}{|\bar a \times \bar b|}$ = $\displaystyle -i + j + k$..............Is this correct ??
Yes, it is. But whether or not it got full marks would depend on whether the person marking it wanted a simplified answer. Certainly at this level the common factor of 5 should be cancelled. It is personal taste whether or not to rationalise the denominator. You should discuss this further with your instructor.