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  4. The unit vector perpendicular to vectors hat i + hatj and hati - hatj forming a right handed system is

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Q. The unit vector perpendicular to vectors $\hat {i} + \hat{j} $ and $ \hat{i} - \hat{j} $ forming a right handed system is

AMUAMU 2013Vector Algebra

Solution:

Let the required vector be $a$. Then, $I+J, I-J$ and $a$ form a right handed system
$\therefore a =( i + j ) \times( i - j ) $
$=\begin{vmatrix}i & j & k \\1 & 1 & 0 \\1 & -1 & 0\end{vmatrix} $
$=i(0-0)-j(0-0)+k(-1-1) $
$=-2 k$
Hence, the required unit vector is
$a =\frac{ a }{| a |}=\frac{-2 k}{|-2 k|}=\frac{-2 k}{\sqrt{4}}=-k$