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  4. A square ABCD is of diagonal 2 a is folded along the diagonal AC so that plane DAC , BAC are at right angles then the shortest distance between DC and AB

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Q. A square $ABCD$ is of diagonal $2 a$ is folded along the diagonal $AC$ so that plane $DAC , BAC$ are at right angles then the shortest distance between $DC$ and $AB$

Vector Algebra

Solution:

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Let $O$ be the centre of square & $O A, O B, O D$ along $x, y, z$ axes respectively $\Rightarrow A =( a , 0,0), B =(0, a , 0)$, $D =(0,0, a ) \& C \equiv(- a , 0,0)$ equation of $AB$ are
$\frac{ x - a }{ a }=\frac{ y -0}{- a }=\frac{ z -0}{0}$
and equation of $DC$ are
$\frac{x-0}{a}=\frac{y-0}{0}=\frac{z-a}{a}$
Hence,$SD =\frac{2 a }{\sqrt{3}}$