How to Find the Normal Vector of the Second Plane?

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Discussion Overview

The discussion revolves around finding the normal vector of a second plane given the normal vector of a first plane, the angle between the two planes, and a point on their line of intersection. The inquiry seeks to explore the mathematical relationships and conditions that would allow for the determination of the second plane's normal vector.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant states that the angle between the planes being 60 degrees implies that the angle between their normal vectors is also 60 degrees, leading to an infinite number of possible normal vectors for the second plane.
  • Another participant suggests that knowing two points on the line of intersection could potentially help in deriving the normal of the second plane.
  • A further contribution indicates that the normals of the planes are perpendicular to the line of intersection, which could provide additional equations involving the components of the normal vector of the second plane.
  • One participant proposes using geometric relationships to derive a third equation involving the normal vector components based on the angles in a quadrilateral formed by the planes.

Areas of Agreement / Disagreement

Participants generally agree that the information provided is insufficient to uniquely determine the normal vector of the second plane. Multiple competing views exist regarding the conditions and additional information that could aid in solving the problem.

Contextual Notes

Limitations include the lack of specific equations for the planes and the dependence on the geometric interpretation of the angles involved. The discussion does not resolve the mathematical steps necessary to find the normal vector.

the_rising
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Hello everyone,

I posted this question before also but did not get the satisfactory answer. So I am posting the question again with more specifications. My question is --- Suppose we have two planes intersecting in one line. Let one plane's normal be [3,4,2] and other plane's normal be [a,b,c]. We know the angle between the two planes as 60 degrees. We know one point on the intersecting line, suppose (7,3,4). Now with this information in hand, can we find out the normal vector of second plane ie [a,b,c].
All the responses are welcome.

Regards,
 
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This is not enough information. If the angle between the planes is 60 degrees then the angle between normal vectors is also 60 degrees. But there are an infinite number of (unit) vectors that make a 60 degree angle with [3,4,2]. Any vector on the cone making a 60 degree angle with [3,4,2] as its axis satisfies that.
 
HallsofIvy said:
This is not enough information. If the angle between the planes is 60 degrees then the angle between normal vectors is also 60 degrees. But there are an infinite number of (unit) vectors that make a 60 degree angle with [3,4,2]. Any vector on the cone making a 60 degree angle with [3,4,2] as its axis satisfies that.

--------

Thanks for the info...
If we know two points on the line of intersection, then is it possible to find out the normal of the second plane ?
 
If the planes make angle of 60 degrees at the point of intersection , then the normals also make the same angle, this condition will give you an equation in a,b and c.
Secondly, the normal of any of the two planes will be prependicular to the line of intersection of two planes.You can find the line of intersection by subtracting one plane equation from other.This will give you one more equation in a,b and c.If the equation of planes is not given , use the point on the intersection-line given .

Now make a quadrilateral of which two angle are 90 degree, one is 60 degree , find the fourth one , using this and some geometry find the length of the normal (for which u need the a,b,c) , this gives you the third equation.Solve the three variables.

BJ
 
the_rising said:
--------

Thanks for the info...
If we know two points on the line of intersection, then is it possible to find out the normal of the second plane ?

If you know two points on the line of intersection of two planes , you can derive the equation of the line , which will make your job easier.

BJ
 

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