Line of intersection of two planes

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Discussion Overview

The discussion revolves around understanding why the cross product of the normal vectors of two planes yields the direction vector of their line of intersection. It explores the geometric relationships between the planes and their normals.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant expresses difficulty in understanding the relationship between the cross product of normal vectors and the line of intersection.
  • Another participant explains that the line of intersection lies in both planes and that the normal to a plane is perpendicular to any line in that plane, suggesting that the cross product results in a direction parallel to the line of intersection.
  • A similar point is reiterated, emphasizing that the cross product gives a line perpendicular to both normals, thus lying in both planes and being parallel to the line of intersection.
  • A further clarification is made that the cross product is a vector with direction but no specific position, indicating that while it is parallel to the line of intersection, it does not lie on it.

Areas of Agreement / Disagreement

Participants generally agree on the geometric interpretation of the cross product in relation to the line of intersection, but there is some contention regarding the precise nature of the cross product as a vector versus a line.

Contextual Notes

The discussion does not resolve the nuances of the definitions involved, particularly regarding the nature of vectors and lines in space.

VenaCava
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Hi,
I am having difficutly figuring out why the cross product of the normal vectors of each plane gives the direction vector of the line of intersection. Anyone care to try to explain?


Thanks!
 
Mathematics news on Phys.org
The line of intersection lies in both planes. The normal to a plane is (by definition) perpendicular to any line in the plane. The cross product then gives you a line perpendicular to both normals, so that it must be parallel to the line of intersection.
 
mathman said:
The line of intersection lies in both planes. The normal to a plane is (by definition) perpendicular to any line in the plane. The cross product then gives you a line perpendicular to both normals,
therefore lieing in both planes, therefore along the line of intersection
so that it must be parallel to the line of intersection.
 
therefore lieing in both planes, therefore along the line of intersection

The cross product is a vector, NOT a line is space - that is, it has a direction but no position. Therefore it doesn't lie anywhere, but it is parallel to the line of intersection.
 

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