Let the vectors $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ and…
Let the vectors $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ and $\overline{\mathrm{d}}$ be such that $(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times(\overline{\mathrm{c}} \times \overline{\mathrm{d}})=\overline{0}$. Let $\mathrm{P}_1$ and $\mathrm{P}_2$ be the planes determined by the pair of vectors $\stackrel{\rightharpoonup}{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}, \overline{\mathrm{d}}$ respectively, then the angle between $P_1$ and $P_2$ is
0
$\frac{\pi}{4}$
$\frac{\pi}{3}$
$\frac{\pi}{2}$
Solution
According to the given condition, we get Normal to the plane $P_1$ is parallel to $\overline{\mathrm{a}} \times \overline{\mathrm{b}}$ and normal to the plane $\mathrm{P}_2$ is parallel to $\overline{\mathrm{c}} \times \overline{\mathrm{d}}$.
Given that $(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times(\overline{\mathrm{c}} \times \overline{\mathrm{d}})=0$
$\therefore \quad(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \|(\overline{\mathrm{c}} \times \overline{\mathrm{d}})$
$\therefore \quad$ Angle between $P_1$ and $P_2$ is 0 .