Question

Two planes are perpendicular to each other,one of them contains vector $\overline{a}$ and $\overline{b}$, other contains $\overline{c}$ and $\overline{d}$ then $(\overline{a}\times \overline{b})\cdot (\overline{c}\times \overline{d})=$

• A. $1$
• B. $0$
• C. $\left[\overline{a}\overline{b}\overline{c}\right]$
• D. $\left[\overline{b}\overline{c}\overline{d}\right]$

Answer 1

Answer: (B)

$0$

Normal vector to plane $\overline{a},\overline{b}$ is given by, $\overline{a}\times \overline{b}$
And normal vector to plane $\overline{c},\overline{d}$ is given by, $\overline{c}\times \overline{d}$
Now for planes to be pependicular, the angle between their normal should be ${90}^{\circ }$
$\Rightarrow (\overline{a}\times \overline{b})\cdot (\overline{c}\times \overline{d})=|\overline{a}\times \overline{b}||\overline{c}\times \overline{d}|\cos {90}^{\circ }=0$