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

We have $\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{b}\times \overrightarrow{d}\Rightarrow \overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{d})$

$\Rightarrow (\overrightarrow{a}\cdot \overrightarrow{c})\overrightarrow{b}-(\overrightarrow{a}\cdot \overrightarrow{b})\overrightarrow{c}=(\overrightarrow{a}\cdot \overrightarrow{d})\overrightarrow{b}-(\overrightarrow{a}\cdot \overrightarrow{b})\overrightarrow{d}(\because \overrightarrow{a}\cdot \overrightarrow{d}=0)$

$\Rightarrow (\overrightarrow{a}\cdot \overrightarrow{b})\overrightarrow{d}=-(\overrightarrow{a}\cdot \overrightarrow{c})\overrightarrow{b}+(\overrightarrow{a}\cdot \overrightarrow{b})c\Rightarrow \overrightarrow{d}=\overrightarrow{c}-\left(\frac{\overrightarrow{a}\cdot \overrightarrow{c}}{\overrightarrow{a}\cdot \overrightarrow{b}}\right)\overrightarrow{b}$