Question

A unit vector perpendicular to the plane containing the vectors $\hat {i}-\hat {j}+\hat{k}$ and $-\hat{i}+\hat {j}+\hat{k}$ is

• A. $\pm \frac {\hat {i}-\hat {j}} {\sqrt {2}}$
• B. $\pm \frac {\hat {i}-\hat {k}} {\sqrt {2}}$
• C. $\pm \frac {\hat {j}-\hat {k}} {\sqrt {2}}$
• D. $\mp \frac {\hat {i}+\hat {j}} {\sqrt {2}}$

Answer 1

Answer: (D)

$\mp \frac {\hat {i}+\hat {j}} {\sqrt {2}}$

Answer 2

Let $a =\hat{ i }-\hat{ j }+\hat{ k }, b =-\hat{ i }+\hat{ j }+\hat{ k }$ An unit vector perpendicular to the plane a and b $=\pm \frac{ a \times b }{| a \times b |}$ Now, $a \times b =\begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\\ 1 & -1 & 1 \\\ -1 & 1 & 1\end{vmatrix}$ $=\hat{ i }(-1-1)-\hat{ j }(1+1)+\hat{ k }(1-1)$ $=-2 \hat{ i }-2 \hat{ j }$ $| a \times b |=\sqrt{2^{2}+2^{2}}=2 \sqrt{2}$ $\therefore \frac{ a \times b }{| a \times b |} =\mp \frac{2(\hat{ i }+\hat{ j })}{2 \sqrt{2}}=\mp \frac{(\hat{ i }+\hat{ j })}{\sqrt{2}}$