Find an equation of the plane passing through the three points given. P = (2, 0, 0), Q = (0, 4, 0), R = (0, 0, 2)

Question
Analytic geometry
Find an equation of the plane passing through the three points given. P = (2, 0, 0), Q = (0, 4, 0), R = (0, 0, 2)

2021-03-07
Let P=(2,0,0) Q=(0,4,0) and R=(0,0,2)
First, find a normal vector n
$$n=\vec{PQ}\cdot\vec{PR}$$
Therefore,
$$\vec{PQ}=(0,4,0)-(2,0,0)=(-2,4,0)$$
$$\vec{PR}=(0,0,2)-(2,0,0)=(-2,0,2)$$
$$n=\vec{PQ}\cdot\vec{PR}=\begin{bmatrix}i&j&k\\-2&4&0\\-2&0&2\end{bmatrix}$$
$$n=\begin{bmatrix}4&0\\0&2\end{bmatrix}i-\begin{bmatrix}-2&0\\-2&2\end{bmatrix}j+\begin{bmatrix}-2&4\\-2&0\end{bmatrix}k$$
$$n=(8-0)i-(-4+0)j+(0+8)k$$
$$n=8i+4j+8k$$
$$n=(8,4,8)$$
We have the equation 8x+4y+8z=d
Now choose any one of the three points, using P=(2,0,0)
$$d=n\cdot\vec{OP}$$
$$d=(8,4,8)\cdot(2,0,0)$$
d=16
d=1
Therefore
8x+4y+8z=16
2x+y+2z=4
Results
2x+y+2z=4

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