Question
Which one of the following is the unit vector perpendicular to both $$\overrightarrow a = - \hat i + \hat j + \hat k$$ and $$\overrightarrow b = \hat i - \hat j + \hat k\,?$$
A.
$$\frac{{\hat i + \hat j}}{{\sqrt 2 }}$$
B.
$${\hat k}$$
C.
$$\frac{{\hat j + \hat k}}{{\sqrt 2 }}$$
D.
$$\frac{{\hat i - \hat j}}{{\sqrt 2 }}$$
Answer :
$$\frac{{\hat i + \hat j}}{{\sqrt 2 }}$$
Solution :
According to question
$$\overrightarrow a = - \hat i + \hat j + \hat k$$ and $$\overrightarrow b = \hat i - \hat j + \hat k$$
Then, \[a \times b = \left| \begin{array}{l}
\,\,\,\hat i\,\,\,\,\,\,\,\hat j\,\,\,\,\,\,\,\,\hat k\\
- 1\,\,\,\,\,\,1\,\,\,\,\,\,1\\
\,\,\,1\,\, - 1\,\,\,\,1
\end{array} \right|\]
$$\eqalign{
& = \hat i\left[ {1 + 1} \right] - \hat j\left[ { - 1 - 1} \right] + \hat k\left[ {1 - 1} \right] \cr
& = 2\hat i + 2\hat j + 0 \cr
& = 2\left( {\hat i + \hat j} \right) \cr
& {\text{and }}\left| {a \times b} \right| = \sqrt {4 + 4} = 2\sqrt 2 \cr} $$
$$\therefore $$ Required unit vector $$ = \pm \frac{{2\left( {\hat i + \hat j} \right)}}{{2\sqrt 2 }} = \pm \frac{{\hat i + \hat j}}{{\sqrt 2 }}$$