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


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

$Find the equation for the plane$ $through the point A (6, 2, - 4),$ $B (- 2, 4, 8), C (4, - 2, 2) . - Vector Analysis$ $M . m$
	Answered by cortano1 last updated on 18/Dec/22

	$\vec{n} = \vec{A B} \times \vec{A C} = \| \begin{matrix} - 8 2 12 \\ - 2 - 4 6 \end{matrix} \|$ $= (60, 24, 36)$ $\vec{A X} = (x - 6, y - 2, z + 4)$ $e q o f p l a n e \Rightarrow \vec{A X} . \vec{n} = 0$ $\Rightarrow 60 (x - 6) + 24 (y - 2) + 36 (z + 4) = 0$ $\Rightarrow 5 x - 30 + 2 y - 4 + 3 z + 12 = 0$ $\Rightarrow 5 x + 2 y + 3 z - 22 = 0$
	Answered by mr W last updated on 18/Dec/22

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

	$\vec{A B} = (- 8, 2, 12)$ $\vec{A C} = (- 2, - 4, 6)$ $\vec{n} = \vec{A B} \times \vec{A C} = (- 8, 2, 12) \times (- 2, - 4, 6)$ $= (60, 24, 36)$ $s a y a n y p o i n t o n t h e p l a n e i s$ $P (x, y, z)$ $\vec{A P} = (x - 6, y - 2, z + 4)$ $w e h a v e \vec{A P} ⊥ \vec{n},$ $\Rightarrow \vec{A P} \cdot \vec{n} = 0$ $\Rightarrow 60 (x - 6) + 24 (y - 2) + 36 (z + 4) = 0$ $\Rightarrow 10 x + 4 y + 3 z - 22 = 0 ✓$
	Commented by cortano1 last updated on 18/Dec/22

	$w h y \vec{n} = (60, 24, 36) ?$
	Commented by mr W last updated on 18/Dec/22

	$\vec{n} = (- 8, 2, 12) \times (- 2, - 4, 6) = (60, 24, 36)$ $y o u g o t a d i f f e r e n t \vec{n}, b e c a u s e y o u$ $w r o n g l y t o o k \vec{A B} = (- 8, 2, - 12) .$ $p l e a s e r e c h e c k .$
	Commented by cortano1 last updated on 18/Dec/22

	$o o h h h y e s . b u t y o u r a n s w e r h a v e$ $a t y p o$
	Commented by mr W last updated on 18/Dec/22

	$t h a n k s! i h a v e f i x e d i t .$
	Commented by Mastermind last updated on 18/Dec/22

	$Thank you boss$
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