Step by step solution to "Find an equation of the plane passing through..."

SchachtN

Answered question

2021-03-06

Find an equation of the plane passing through the three points given. $P = (2, 0, 0),$

$Q = (0, 4, 0),$

$R = (0, 0, 2)$

Answer & Explanation

broliY

Skilled2021-03-07Added 97 answers

Let $P = (2, 0, 0) Q = (0, 4, 0)$ and $R = (0, 0, 2)$
First, find a normal vector n
$n = \vec{P Q} \cdot \vec{P R}$
Therefore,
$\vec{P Q} = (0, 4, 0) - (2, 0, 0) = (- 2, 4, 0)$
$\vec{P R} = (0, 0, 2) - (2, 0, 0) = (- 2, 0, 2)$
$n = \vec{P Q} \cdot \vec{P R} = [\begin{matrix} i & j & k \\ - 2 & 4 & 0 \\ - 2 & 0 & 2 \end{matrix}]$
$n = [\begin{matrix} 4 & 0 \\ 0 & 2 \end{matrix}] i - [\begin{matrix} - 2 & 0 \\ - 2 & 2 \end{matrix}] j + [\begin{matrix} - 2 & 4 \\ - 2 & 0 \end{matrix}] k$
$n = (8 - 0) i - (- 4 + 0) j + (0 + 8) k$
$n = 8 i + 4 j + 8 k$
$n = (8, 4, 8)$
We have the equation $8 x + 4 y + 8 z = d$
Now choose any one of the three points, using $P = (2, 0, 0)$
$d = n \cdot \vec{O P}$
$d = (8, 4, 8) \cdot (2, 0, 0)$
$d = 16$
$d = 1$
Therefore
$8 x + 4 y + 8 z = 16$
$2 x + y + 2 z = 4$
Results
$2 x + y + 2 z = 4$

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