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

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