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.](Q45856.png)
$${Prove}\:{that}\:{the}\:{length}\:{of}\:{the}\:{perpendicular} \\ $$$${from}\:{the}\:{origin}\:{to}\:{the}\:{plane}\:{passing} \\ $$$${through}\:{point}\:\overset{\rightarrow} {{a}}\:{and}\:{containing}\:{the} \\ $$$${line}\:\overset{\rightarrow} {{r}}=\overset{\rightarrow} {{b}}+\lambda\overset{\rightarrow} {{c}}\:{is}\:\frac{\left[\overset{\rightarrow} {{a}}\:\:\overset{\rightarrow} {{b}}\:\:\overset{\rightarrow} {{c}}\:\right]}{\mid\overset{\rightarrow} {{b}}×\overset{\rightarrow} {{c}}\:+\overset{\rightarrow} {{c}}×\overset{\rightarrow} {{a}}\mid}\:. \\ $$$${Here}\:\left[\overset{\rightarrow} {{a}}\:\overset{\rightarrow} {{b}}\:\overset{\rightarrow} {{c}}\right]\:=\:{scalar}\:{triple}\:{product}. \\ $$
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^→ ]....](Q45857.png)
$${I}'{ve}\:{got}\:{equation}\:{of}\:{plane}\:{as}: \\ $$$$\overset{\rightarrow} {{r}}.\left(\overset{\rightarrow} {{b}}×\overset{\rightarrow} {{c}}+\overset{\rightarrow} {{c}}×\overset{\rightarrow} {{a}}\right)=\:\left[\overset{\rightarrow} {{a}}\:\overset{\rightarrow} {{b}}\:\overset{\rightarrow} {{c}}\:\right].... \\ $$