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Prove-that-the-length-of-the-perpendicular-from-the-origin-to-the-plane-passing-through-point-a-and-containing-the-line-r-b-c-is-a-b-c-b-c-c-a-Here-a




Question Number 45856 by rahul 19 last updated on 17/Oct/18
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.
$${Prove}\:{that}\:{the}\:{length}\:{of}\:{the}\:{perpendicular} \\ $$$${from}\:{the}\:{origin}\:{to}\:{the}\:{plane}\:{passing} \\ $$$${through}\:{point}\:\overset{\rightarrow} {{a}}\:{and}\:{containing}\:{the} \\ $$$${line}\:\overset{\rightarrow} {{r}}=\overset{\rightarrow} {{b}}+\lambda\overset{\rightarrow} {{c}}\:{is}\:\frac{\left[\overset{\rightarrow} {{a}}\:\:\overset{\rightarrow} {{b}}\:\:\overset{\rightarrow} {{c}}\:\right]}{\mid\overset{\rightarrow} {{b}}×\overset{\rightarrow} {{c}}\:+\overset{\rightarrow} {{c}}×\overset{\rightarrow} {{a}}\mid}\:. \\ $$$${Here}\:\left[\overset{\rightarrow} {{a}}\:\overset{\rightarrow} {{b}}\:\overset{\rightarrow} {{c}}\right]\:=\:{scalar}\:{triple}\:{product}. \\ $$
Commented by rahul 19 last updated on 17/Oct/18
I′ve got equation of plane as:  r^→ .(b^→ ×c^→ +c^→ ×a^→ )= [a^→  b^→  c^→  ]....
$${I}'{ve}\:{got}\:{equation}\:{of}\:{plane}\:{as}: \\ $$$$\overset{\rightarrow} {{r}}.\left(\overset{\rightarrow} {{b}}×\overset{\rightarrow} {{c}}+\overset{\rightarrow} {{c}}×\overset{\rightarrow} {{a}}\right)=\:\left[\overset{\rightarrow} {{a}}\:\overset{\rightarrow} {{b}}\:\overset{\rightarrow} {{c}}\:\right]…. \\ $$

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