Let $$\overrightarrow a = \overrightarrow i + \overrightarrow j + \overrightarrow k ,\,\overrightarrow c = \overrightarrow j - \overrightarrow k .$$       If $$\overrightarrow b $$ is a vector satisfying $$\overrightarrow a \times \overrightarrow b = \overrightarrow c $$   and $$\overrightarrow a .\overrightarrow b = 3$$   then $$\overrightarrow b $$ is :

Solution :
$$\eqalign{ & {\text{Let }}\overrightarrow b = x\overrightarrow i + y\overrightarrow j + z\overrightarrow k \cr & \overrightarrow a .\overrightarrow b = 3\,\,\,\, \Rightarrow x + y + z = 3 \cr & \overrightarrow a \times \overrightarrow b = \overrightarrow c \,\,\, \Rightarrow \left( {\overrightarrow i + \overrightarrow j + \overrightarrow k } \right) \times \left( {x\overrightarrow i + y\overrightarrow j + z\overrightarrow k } \right) = \overrightarrow j - \overrightarrow k \cr & {\text{or }}\left( {z - y} \right)\overrightarrow i + \left( {x - z} \right)\overrightarrow j + \left( {y - x} \right)\overrightarrow k = \overrightarrow j - \overrightarrow k \cr & \Rightarrow \,z - y = 0,\,x - z = 1,\,y - x = - 1 \cr} $$
Solving the four equations in $$x,\,y,\,z,$$   we get $$x = \frac{5}{3},\,y = \frac{2}{3},\,z = \frac{2}{3}.$$