Prove that the length of the perpendicular from the origin to the plane passing through point a^→ and containing the line r^→ =b^→ +λc^→ is (([a^→ b^→ c^→ ])/(∣b^→ ×c^→ +c^→ ×a^→ ∣)) . Here [a^→ b^→ c^→ ] = scalar triple product.


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Question Number 45856 by rahul 19 last updated on 17/Oct/18

$P r o v e t h a t t h e l e n g t h o f t h e p e r p e n d i c u l a r$ $f r o m t h e o r i g i n t o t h e p l a n e p a s s i n g$ $t h r o u g h p o i n t \vec{a} a n d c o n t a i n i n g t h e$ $l i n e \vec{r} = \vec{b} + λ \vec{c} i s \frac{[\vec{a} \vec{b} \vec{c}]}{∣ \vec{b} \times \vec{c} + \vec{c} \times \vec{a} ∣} .$ $H e r e [\vec{a} \vec{b} \vec{c}] = s c a l a r t r i p l e p r o d u c t .$
Commented by rahul 19 last updated on 17/Oct/18

$I^{'} v e g o t e q u a t i o n o f p l a n e a s :$ $\vec{r} . (\vec{b} \times \vec{c} + \vec{c} \times \vec{a}) = [\vec{a} \vec{b} \vec{c}] . . . .$
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