I want to know which of the two spaces each point belongs to in a 3-dimensional space divided into two parts by a single plane.

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Yusaku Ohta
Yusaku Ohta 2023-8-30
评论： Yusaku Ohta 2023-8-30
Suppose there are multiple points in a 3-dimensional space.
If the 3-dimensional space is divided into two by one plane,
We want to know which of the two spaces each point belongs to.
Let the plane be a(x-p)+b(y-q)+c(z-r)=0.
In this case, the point on the plane is P(x,y,z).
Please tell me how to find it.
Thank you in advance.

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采纳的回答

Torsten
Torsten 2023-8-30
编辑：Torsten 2023-8-30
The two half-spaces that IR^3 is divided in by the plane are a(x-p)+b(y-q)+c(z-r) < 0 and a(x-p)+b(y-q)+c(z-r) > 0.
So for a given point P = (x,y,z), you have to check whether a(x-p)+b(y-q)+c(z-r) < 0 or a(x-p)+b(y-q)+c(z-r) > 0.
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recent works
recent works 2023-8-30
The two half-spaces that IR^3 is divided in by the plane are a(x-p)+b(y-q)+c(z-r) < 0 and a(x-p)+b(y-q)+c(z-r) > 0.
So for a given point P = (x,y,z), you have to check whether a(x-p)+b(y-q)+c(z-r) < 0 or a(x-p)+b(y-q)+c(z-r) > 0.
If the result is less than zero, then the point is in the half-space where a(x-p)+b(y-q)+c(z-r) < 0.
If the result is greater than zero, then the point is in the half-space where a(x-p)+b(y-q)+c(z-r) > 0.
If the result is equal to zero, then the point is on the plane.

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更多回答（1 个）

recent works
recent works 2023-8-30
To find which of the two spaces each point belongs to, you can use the following steps:
  1. Substitute the coordinates of the point into the equation of the plane.
  2. If the result is zero, then the point is on the plane.
  3. If the result is not zero, then the point is not on the plane.
For example, let's say we have the following plane:
a(x-p)+b(y-q)+c(z-r)=0
and the following point:
P(x,y,z)=(1,2,3)
To find which space this point belongs to, we would substitute the coordinates of the point into the equation of the plane.
a(1-p)+b(2-q)+c(3-r)=0
If the result is zero, then the point is on the plane. Otherwise, the point is not on the plane.
In this case, the result is not zero, so the point (1,2,3) is not on the plane.
keep in mind:
  • The equation of the plane can be written in different ways. For example, it can also be written as:
(x-p)/a=(y-q)/b=(z-r)/c=k
where k is a constant.
  • The point on the plane can be any point that satisfies the equation of the plane.
  • The two spaces divided by the plane are called the half-spaces.
  • The point that is exactly on the plane belongs to both half-spaces.

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