Question

The unit vector perpendicular to the vectors $\hat i - \hat j$ and $\hat i + \hat j$ forming a right-handed system is?
(A) $\hat k$
(B) $ - \hat k$
(C) $\dfrac{{\hat i + \hat j}}{{\sqrt 2 }}$
(D) $\dfrac{{\hat i - \hat j}}{{\sqrt 2 }}$

Answer 1

Hint: In order to solve this question, we should know that the vector product of any two vectors produces another vector that is perpendicular to both the vectors and hence forms a right-handed system, here we will find the unit vector perpendicular to the given two vectors using vector product.

Formula Used:
 f we have two vectors in component form as $\vec a = x\hat i + y\hat j + z\hat k$ and $\vec b = x'\hat i + y'\hat j + z'\hat k$ then, vector product between two vectors is calculated as
\[\vec a \times \vec b = \left| {\begin{array}{*{20}{c}}
  {\hat i}&{\hat j}&{\hat k} \\
  x&y&z \\
  {x'}&{y'}&{z'}
\end{array}} \right|\]

Complete step by step Solution:
We have given the two vectors $\hat i - \hat j$ and $\hat i + \hat j$, let us name them
$
  \vec a = \hat i - \hat j \\
  \vec b = \hat i + \hat j \\
 $ now, to find the third vector which is perpendicular to both we will find the vector product between these two vectors using the formula for two general vectors as \[\vec a \times \vec b = \left| {\begin{array}{*{20}{c}}
  {\hat i}&{\hat j}&{\hat k} \\
  x&y&z \\
  {x'}&{y'}&{z'}
\end{array}} \right|\] where, $x,y,z\;\& \;x',y',z'$ are the components of the vector so, on putting the values we get,
\[
  \vec a \times \vec b = \left| {\begin{array}{*{20}{c}}
  {\hat i}&{\hat j}&{\hat k} \\
  1&{ - 1}&0 \\
  1&1&0
\end{array}} \right| \\
  \vec a \times \vec b = \hat i(0) - \hat j(0) + \hat k(2) \\
  \vec a \times \vec b = 2\hat k \\
 \]
So, the vector perpendicular to both the vectors $\hat i - \hat j$ and $\hat i + \hat j$ is\[2\hat k\] now, converting this vector into the unit vector we get,
$
  \dfrac{{2\hat k}}{{\sqrt {{2^2}} }} \\
   = \hat k \\
 $
So, a unit vector perpendicular to both the vectors $\hat i - \hat j$ and $\hat i + \hat j$ is $\hat k$

Hence, the correct option is A.

Note: It should be remembered that while calculating the determinant of the cross product of two vectors always make sure the sign of each term is determined correctly and cross-product is one of the kinds of vectors product other than scalar product and triple product.