The vectors $${\vec a}$$ and $${\vec b}$$ are not perpendicular and $${\vec c}$$ and $${\vec d}$$ are two vectors satisfying $$\vec b \times \vec c = \vec b \times \vec d$$   and $$\vec a.\vec d = 0$$   Then the vector $${\vec d}$$ is equal to :

Solution :
$$\eqalign{ & \vec a.\vec b \ne 0,\,\,\,\vec a.\vec d = 0 \cr & {\text{Now, }}\vec b \times \vec c = \vec b \times \vec d \cr & \Rightarrow \vec a \times \left( {\vec b \times \vec c} \right) = \vec a \times \left( {\vec b \times \vec d} \right) \cr & \Rightarrow \left( {\vec a.\vec c} \right)\vec b - \left( {\vec a.\vec b} \right)\vec c = \left( {\vec a.\vec d} \right)\vec b - \left( {\vec a.\vec b} \right)\vec d \cr & \Rightarrow \left( {\vec a.\vec b} \right)\vec d = - \left( {\vec a.\vec c} \right)\vec b + \left( {\vec a.\vec b} \right)\vec c \cr & \Rightarrow \vec d = \vec c - \left( {\frac{{\vec a.\vec c}}{{\vec a.\vec b}}} \right)\vec b \cr} $$