The vectors \(\bar{a}\) and \(\bar{b}\) are not perpendicular and \(\overline{\mathrm{c}}\) and \(\overline{\mathrm{d}}\) are two vectors satisfying \(\overline{\mathrm{b}} \times \overline{\mathrm{c}}=\overline{\mathrm{b}} \times \overline{\mathrm{d}}\) and \(\overline{\mathrm{a}} \cdot \overline{\mathrm{d}}=0\), then the vector \(\overline{\mathrm{d}}\) is equal to

Given,
Vectors \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) are not perpendicular
\(\begin{aligned}
\therefore \quad & \overline{\mathrm{a}} \cdot \overline{\mathrm{~b}} \neq 0 \\
& \overline{\mathrm{a}} \cdot \overline{\mathrm{~d}}=0 \\
& \overline{\mathrm{~b}} \times \overline{\mathrm{c}}=\overline{\mathrm{b}} \times \overline{\mathrm{d}} \\
& \Rightarrow \overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})=\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{d}}) \\
& \Rightarrow(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{b}}-(\overline{\mathrm{a}} \cdot \overline{\mathrm{~b}}) \overline{\mathrm{c}}=(\overline{\mathrm{a}} \cdot \overline{\mathrm{~d}}) \overline{\mathrm{b}}-(\overline{\mathrm{a}} \cdot \overline{\mathrm{~b}}) \overline{\mathrm{d}} \\
& \Rightarrow(\overline{\mathrm{a}} \cdot \overline{\mathrm{~b}}) \overline{\mathrm{d}}=-(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}) \cdot \overline{\mathrm{b}}+(\overline{\mathrm{a}} \cdot \overline{\mathrm{~b}}) \overline{\mathrm{c}} \\
\quad & \Rightarrow \overline{\mathrm{~d}}=\frac{-(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{b}}}{\overline{\mathrm{a}} \cdot \overline{\mathrm{~b}}}+\overline{\mathrm{c}} \\
\Rightarrow & \overline{\mathrm{~d}}=\overline{\mathrm{c}}-\left(\frac{\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}}{\overline{\mathrm{a}} \cdot \overline{\mathrm{~b}}}\right) \overline{\mathrm{b}}
\end{aligned}\)