The vector having magnitude equal to $$3$$ and perpendicular to the two vectors $$\vec A = 2\hat i + 2\hat j + \hat k$$    and $$\vec B = 2\hat i - 2\hat j + 3\hat k$$    is:

Solution :
The required vector is,
$$\eqalign{ & = 3\frac{{\left( {\vec A \times \vec B} \right)}}{{\left| {\vec A \times \vec B} \right|}} = 3\frac{{\left[ {\left( {2\hat i + 2\hat j + \hat k} \right) \times \left( {2\hat i - 2\hat j + 3\hat k} \right)} \right]}}{{\left| {\vec A \times \vec B} \right|}} \cr & = 3\frac{{\left( {8\hat i - 4\hat j - 8\hat k} \right)}}{{\sqrt {{8^2} + {4^2} + {8^2}} }} \cr & = 2\hat i - \hat j - 2\hat k. \cr} $$