MHB What is the Maximum Inclination of a 3D Plane and its Equation in the XY Plane?

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The maximum inclination of a 3D plane represented by the equation Ax + By + Cz = 0 is achieved when it is perpendicular to the intersection line with the reference plane. The slope of this maximum inclination line can be determined by taking the negative reciprocal of the slope of the intersection line, given as -1/m. If the intersection line is horizontal (slope m = 0), the perpendicular line is vertical and does not have a defined slope. Understanding these relationships is crucial for accurately determining the inclination in the XY plane. The discussion clarifies the geometric principles involved in these calculations.
cptolemy
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Hi,

If I have a 3D plane Ax+By+Cz=0, and I know the line equation of its intersection with the reference plane, what is the maximum inclination a point can have and what is its equation in the xy plane?

Is it perpendicular to the intersection? That is , if the slope of the intersection line is m, the max will be given by the line with a slope of -1/m?

This is embarassing...

Cheers,

Kepler
 
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If I understand what you mean by "inclination" then, yes, the plane with maximum "inclination" to a given plane is perpendicular to it. And, yes, if a line is perpendicular to a line with (non-zero) slope m, then it has slope -1/m. Note that if m= 0, so "horizontal", then the perpendicular line is "vertical" and does not have a defined slope.
 

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