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Distance between Two Parallel Planes

If two lines ...

Question

If two lines joining the pair of points $(4, 1, 2)$ , $(p, 3, 0)$ , $(2, 1, - 1)$ , and $(4, 3, 2)$ are mutually perpendicular to each other then find the value of $p$ .

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Solution

$\begin{matrix} A (4, 1, 2), B (p, 3, 0) L i n e j o i n i n g A a n d B i s p a r a l l e l t o 4 - p, 1 - 3, 2 - 0 (4 - p, - 2, 2) C (2, 1, - 1) D (4, 3, 2) L i n e j o i n i n g C a n d D i s p a r a l l e l t o 2 - 4, 1 - 3, - 1 - 2 (- 2, - 2, - 3) - 2 (4 - p) - 2 (- 2) + 2 (- 3) = 0 - 8 + 2 p + 4 - 6 = 0 2 p = 0 p = 5 \end{matrix}$

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