Find the normal form of the equation?

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Homework Help Overview

The discussion revolves around finding the normal form of the equation of a plane given a specific point and a normal vector. The subject area is primarily geometry, focusing on planes in three-dimensional space.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the relationship between the normal vector and the equation of the plane, questioning how the constant term is derived when the plane contains a specific point.

Discussion Status

Participants are actively discussing the derivation of the equation of the plane, with some providing insights into the relationship between the normal vector and the point on the plane. There is an exploration of how to ensure the point lies on the plane defined by the normal vector.

Contextual Notes

There is a mention of the general form of the plane equation and the need to find a specific constant that allows the plane to pass through the given point. The discussion reflects an understanding of the geometric implications of parallel planes.

Math9999

Homework Statement


Find the normal form of the equation of the plane that contains the point P=(0, 1, 0) and has normal vector n=[3, 2, 1].

Homework Equations


None.

The Attempt at a Solution


I know that the general form: ax+by=c since that's written in my textbook. But why is the answer 3x+2y+z=2? I know that a=3, b=2 and c=1 based on the normal vector, but how did they get the 2 on the other side?
 
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The equation ##\vec{n} \cdot \vec{x} = 3x+2y+z## gives a plane through the origin with the given normal vector. But we are looking for a parallel plane of it, which contains the given point ##\vec{p}##. So parallel means ##3x+2y+z=c## for some constant ##c##. Now what does it mean for ##c##, that ##\vec{p}=(0,1,0)## satisfies this equation?
 
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So you're saying that c=n*p where 3*0+2*1+1*0=2, and c=2?
 
Yep. This way you make sure that the point is in the plane and it's orientation in space isn't changed.
 
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Thank you!
 

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