Question

Prove that if in tetrahedron , two pair of opposite edges are orthogonal, so is the third pair.

Answer 1

Let ABCD is tetrahedron such that AB$\perp$DC, AC$\perp$BD.

We need to prove AD$\perp$BC.

Let $\overline{DA}=\overline{a},\overline{DB}=\overline{b},\overline{DC}=\overline{c}$

Thus $\overline{c}\cdot (\overline{a}-\overline{b})=0$

and $\overline{b}\cdot (\overline{a}-\overline{c})=0$

which gives

$\overline{a}\cdot \overline{c}=\overline{b}\cdot \overline{c}$

$\Rightarrow \overline{a}\cdot \overline{c}=\overline{a}\cdot \overline{b}$

$\overline{a}\cdot (\overline{b}\cdot \overline{c})=0$

$\therefore AD\perp BC$.