Find-the-equation-for-the-plane-through-the-point-A-6-2-4-B-2-4-8-C-4-2-2-Vector-Analysis-M-m-

Question Number 182984 by Mastermind last updated on 18/Dec/22

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{for}\:\mathrm{the}\:\mathrm{plane} \\ $$$$\mathrm{through}\:\mathrm{the}\:\mathrm{point}\:\mathrm{A}\left(\mathrm{6},\:\mathrm{2},\:−\mathrm{4}\right), \\ $$$$\mathrm{B}\left(−\mathrm{2},\:\mathrm{4},\:\mathrm{8}\right),\:\mathrm{C}\left(\mathrm{4},\:−\mathrm{2},\:\mathrm{2}\right).\:−\mathrm{Vector}\:\mathrm{Analysis} \\ $$$$ \\ $$$$\mathrm{M}.\mathrm{m} \\ $$

Answered by cortano1 last updated on 18/Dec/22

n^(→) = AB^(→) × AC^(→) = determinant (((−8 2 12)),((−2 −4 6))) = (60, 24, 36 ) AX^(→) = (x−6, y−2, z+4) eq of plane ⇒ AX^(→) . n^(→) = 0 ⇒60(x−6)+24(y−2)+36(z+4)=0 ⇒5x−30+2y−4+3z+12=0 ⇒5x+2y+3z−22=0

$$\:\overset{\rightarrow} {{n}}\:=\:\overset{\rightarrow} {{AB}}\:×\:\overset{\rightarrow} {{AC}}\:=\:\begin{vmatrix}{−\mathrm{8}\:\:\:\:\:\:\:\mathrm{2}\:\:\:\:\:\:\:\:\:\:\mathrm{12}}\\{−\mathrm{2}\:\:\:\:−\mathrm{4}\:\:\:\:\:\:\:\:\:\mathrm{6}}\end{vmatrix} \\ $$$$\:=\:\left(\mathrm{60},\:\mathrm{24},\:\mathrm{36}\:\right) \\ $$$$\:\overset{\rightarrow} {{AX}}\:=\:\left({x}−\mathrm{6},\:{y}−\mathrm{2},\:{z}+\mathrm{4}\right) \\ $$$$\:{eq}\:{of}\:{plane}\:\Rightarrow\:\overset{\rightarrow} {{AX}}\:.\:\overset{\rightarrow} {{n}}\:=\:\mathrm{0} \\ $$$$\Rightarrow\mathrm{60}\left({x}−\mathrm{6}\right)+\mathrm{24}\left({y}−\mathrm{2}\right)+\mathrm{36}\left({z}+\mathrm{4}\right)=\mathrm{0} \\ $$$$\Rightarrow\mathrm{5}{x}−\mathrm{30}+\mathrm{2}{y}−\mathrm{4}+\mathrm{3}{z}+\mathrm{12}=\mathrm{0} \\ $$$$\Rightarrow\mathrm{5}{x}+\mathrm{2}{y}+\mathrm{3}{z}−\mathrm{22}=\mathrm{0}\: \\ $$$$ \\ $$

Answered by mr W last updated on 18/Dec/22

Commented by mr W last updated on 18/Dec/22

AB^(→) =(−8, 2, 12) AC^(→) =(−2, −4, 6) n^→ =AB^(→) ×AC^(→) =(−8,2,12)×(−2,−4,6) =(60,24,36) say any point on the plane is P(x,y,z) AP^(→) =(x−6, y−2, z+4) we have AP^(→) ⊥n^(→) , ⇒AP^(→) ∙n^(→) =0 ⇒60(x−6)+24(y−2)+36(z+4)=0 ⇒10x+4y+3z−22=0 ✓

$$\overset{\rightarrow} {{AB}}=\left(−\mathrm{8},\:\mathrm{2},\:\mathrm{12}\right) \\ $$$$\overset{\rightarrow} {{AC}}=\left(−\mathrm{2},\:−\mathrm{4},\:\mathrm{6}\right) \\ $$$$\overset{\rightarrow} {{n}}=\overset{\rightarrow} {{AB}}×\overset{\rightarrow} {{AC}}=\left(−\mathrm{8},\mathrm{2},\mathrm{12}\right)×\left(−\mathrm{2},−\mathrm{4},\mathrm{6}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left(\mathrm{60},\mathrm{24},\mathrm{36}\right) \\ $$$${say}\:{any}\:{point}\:{on}\:{the}\:{plane}\:{is} \\ $$$${P}\left({x},{y},{z}\right) \\ $$$$\overset{\rightarrow} {{AP}}=\left({x}−\mathrm{6},\:{y}−\mathrm{2},\:{z}+\mathrm{4}\right) \\ $$$${we}\:{have}\:\overset{\rightarrow} {{AP}}\bot\overset{\rightarrow} {{n}}, \\ $$$$\Rightarrow\overset{\rightarrow} {{AP}}\centerdot\overset{\rightarrow} {{n}}=\mathrm{0} \\ $$$$\Rightarrow\mathrm{60}\left({x}−\mathrm{6}\right)+\mathrm{24}\left({y}−\mathrm{2}\right)+\mathrm{36}\left({z}+\mathrm{4}\right)=\mathrm{0} \\ $$$$\Rightarrow\mathrm{10}{x}+\mathrm{4}{y}+\mathrm{3}{z}−\mathrm{22}=\mathrm{0}\:\checkmark \\ $$

Commented by cortano1 last updated on 18/Dec/22

$${why}\:\overset{\rightarrow} {{n}}\:=\:\left(\mathrm{60},\mathrm{24},\mathrm{36}\right)? \\ $$

Commented by mr W last updated on 18/Dec/22

n^→ =(−8,2,12)×(−2,−4,6)=(60,24,36) you got a different n^→ , because you wrongly took AB^(→) =(−8,2,−12). please recheck.

$$\overset{\rightarrow} {{n}}=\left(−\mathrm{8},\mathrm{2},\mathrm{12}\right)×\left(−\mathrm{2},−\mathrm{4},\mathrm{6}\right)=\left(\mathrm{60},\mathrm{24},\mathrm{36}\right) \\ $$$${you}\:{got}\:{a}\:{different}\:\overset{\rightarrow} {{n}},\:{because}\:{you} \\ $$$${wrongly}\:{took}\:\overset{\rightarrow} {{AB}}=\left(−\mathrm{8},\mathrm{2},−\mathrm{12}\right). \\ $$$${please}\:{recheck}. \\ $$

Commented by cortano1 last updated on 18/Dec/22