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Q.
If $r$ is a unit vector satisfying $r \times a = b ,| a |=2$ and $| b |=\sqrt{3}$, then one such $r =$
TS EAMCET 2019
Solution:
We have,
$r \times a = b$
$| r \times a |=| b |$
$| r || a | \sin \theta=| b |$
$\sin \theta=\frac{| b |}{| a |}=\frac{\sqrt{3}}{2} \, [\cdot| r |=1,| a |=2,| b |=\sqrt{3}]$
$\therefore \theta=60^{\circ}$
$( r \times a ) \times a = b \times a$
$( r . a ) a -| a |^{2} r = b \times a$
$r =\frac{1}{| a |^{2}}[( a \cdot r ) a - b \times a ]$
$r =\frac{1}{4}((| a \| r | \cos \theta) a - b \times a )$
$\Rightarrow r =\frac{1}{4}( a -( b \times a ))$