Q. Le the vectors $\overrightarrow{\mathbf{a}}\mathbf{,}\overrightarrow{\mathbf{b}}\mathbf{,}\overrightarrow{\mathbf{c}}\$ and $\overrightarrow{\mathbf{d}}$ be such that$\ \left( \overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}} \right)\mathbf{\times}\left( \overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{d}} \right) = 0$. Let $P_{1}$ and $P_{2}$ be planes determined by pair of vectors $\overrightarrow{\mathbf{a}}\mathbf{,}\overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}\mathbf{,}\overrightarrow{\mathbf{d}}$ respectively. Then, the angle between $P_{1}$ and $P_{2}$ is
Explanation
(a)
In plane $P_{1}$, a vector is perpendicular to $\overrightarrow{\mathbf{a}}\ and\ \overrightarrow{\mathbf{b}}$ is $\overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}}$.
In plane $P_{2}$, a vector is perpendicular to $\overrightarrow{\mathbf{c}}$ and $\overrightarrow{\mathbf{d}}$ is $\overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{d}}$
$\Rightarrow \left( \overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}} \right)\mathbf{\times}\left( \overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{d}} \right) = 0$
$\Rightarrow \ \left( \overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}} \right)\mathbf{||}\left( \overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{d}} \right)$
The angle between the planes is 0.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.