The vectors $\overrightarrow a $ and $\overrightarrow b $ are not perpendicular and $\overrightarrow c $ and $\overrightarrow d $ are two vectors satisfying $\overrightarrow b \times \overrightarrow c = \overrightarrow b \times \overrightarrow d $ and $\overrightarrow a .\overrightarrow d = 0\,\,.$ Then the vector $\overrightarrow d $ is equal to :
$\overrightarrow c + \left( {{{\overrightarrow a .\overrightarrow c } \over {\overrightarrow a .\overrightarrow b }}} \right)\overrightarrow b $
$\overrightarrow b + \left( {{{\overrightarrow b .\overrightarrow c } \over {\overrightarrow a .\overrightarrow b }}} \right)\overrightarrow c $
$\overrightarrow c - \left( {{{\overrightarrow a .\overrightarrow c } \over {\overrightarrow a .\overrightarrow b }}} \right)\overrightarrow b $
$\overrightarrow b - \left( {{{\overrightarrow b .\overrightarrow c } \over {\overrightarrow a .\overrightarrow b }}} \right)\overrightarrow c $
