Tetra Perp

Age 16 to 18

Challenge Level Yellow star

A tetrahedron $ABCD$ has vertices $A$ , $B$ , $C$ and $D$ , as shown below:

Show that the edges $AD$ and $BC$ of the tetrahedron are mutually perpendicular if and only if $AB^2+CD^2 = AC^2+BD^2$ .

If the position vector of

$A$ is

${\bf a}$ (and similar for the other vertices), can you use the scalar product to find an expression for

$AB^2$ ?

It might be helpful to note that

$AB^2=|\overrightarrow{AB}|^2$ .

Sometimes it is easier to try and show

$p-q=0$ rather than trying to show

$p=q$ .

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