If $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ lie in a plane, normal to the plane containing $\overline{\mathrm{c}}, \overline{\mathrm{d}}$ then $(\overline{\mathrm{a}}\times \overline{\mathrm{b}})·(\overline{\mathrm{c}}\times \overline{\mathrm{d}})=$

$$\begin{gathered} \text{ The normal of the plane containing }\overrightarrow{a}\text{ and }\overrightarrow{b}\text{ is }=\overrightarrow{a}\times \overrightarrow{b} ···\left(1\right) \\ \text{ The normal of the plane containing }\overrightarrow{\mathrm{c}}\text{ and }\overrightarrow{\mathrm{d}}\text{ is }=\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{d}} ···\left(2\right) \\ \text{ Now the planes are perpendicular to one another. Hence their normal will also } \\ \text{ be perpendicular to one another. } \\ (\overrightarrow{a}\times \overrightarrow{b})\cdot (\overrightarrow{c}\times \overrightarrow{d})=\text{ Null vector } \\ (\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\cdot (\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{d}})=0 \end{gathered}$$