Significance of a plane equation

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    Plane Significance
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Discussion Overview

The discussion revolves around the interpretation of the equation of a plane, specifically the equation x + y + z = 6. Participants explore what the right-hand side (RHS) value of 6 signifies in relation to points on the plane and the implications of this equation in three-dimensional space.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant asserts that if the sum of the coordinates of a point (x, y, z) equals 6, then that point lies on the plane defined by the equation.
  • Another participant questions whether points such as (-50, 0, 56) can be considered on the plane, expressing confusion about the implications of the plane's infinite extent.
  • A participant challenges the initial framing of the equation, suggesting that the coefficients in the equation are fixed and not arbitrary.

Areas of Agreement / Disagreement

Participants generally agree that points satisfying the equation x + y + z = 6 lie on the plane. However, there is some disagreement regarding the interpretation of the equation and the nature of the coefficients.

Contextual Notes

Some assumptions about the nature of the coefficients and the definition of points on the plane remain unresolved, leading to varying interpretations of the equation's implications.

quietrain
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lets say i have a plane equation x + y + z =6

then i know the normal vector is 1 ,1,1 right?

but what does the 6 tells me on the RHS of the equation?

since it is essentially ()x + ()y + ()z, where () is any point on the plane

1)does it mean that i can fit anything into the () as long as i get the total to 6?

like for example 1, 4, 1 gives me 6 too

2)does it then mean that any points i throw in as long as it satisfies 6 will be guaranteed to be a point on that plane?

thanks!
 
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oh i see thank you
 
1 and 2 seem to be saying the same thing. For any point P whose coordinates are (x, y, z), if the sum of the coordinates is 6, P is a point on the plane. If the sum of the coordinates is not 6, P is not on the plane.
 
so i can have (-50 , 0 , 56) as a point on that plane? seems weird :S
 
Why? The plane extends infinitely in all directions. Its intersection with the plane y= 0 is the line x= t, y= 0, z= 6-t, which extends infinitely.

By the way, "()x + ()y + ()z, where () is any point on the plane" plane makes no sense. It looks like you are saying the coefficients can be any numbers but that is not true. It would make more sense to say "()+ ()+ ()= 6".
 
oh i see thanks everyone!
 

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