Intersection of planes / normals question
I'm looking at a further maths question and they determined that since three planes intersect forming a sheaf or triangular prism, the normals to these planes must lie in a plane.
Firstly why is that true? I'm struggling to visualise it.
Also for which other arrangements of planes would this be true?
Firstly why is that true? I'm struggling to visualise it.
Also for which other arrangements of planes would this be true?
Original post by 0-)
I'm looking at a further maths question and they determined that since three planes intersect forming a sheaf or triangular prism, the normals to these planes must lie in a plane.
Firstly why is that true? I'm struggling to visualise it.
Also for which other arrangements of planes would this be true?
Firstly why is that true? I'm struggling to visualise it.
Also for which other arrangements of planes would this be true?
Just imagine a prism with normal vectors coming out of the 3 faces along the body. Where do they lie? It may help to think of each of the intersection lines/edges and how the 2 relevant normals are related, then put the reasoning togethter as the intersection lines/edges are parellel. Alternatively, for a "usual" prism which has a couple of end faces, think of the plane(s) that are defined by the extended end face(s).
Note its the direction of a normal vector thats important. So its really saying that the third normal vector can be written as a linear combination of the other two normals.
(edited 9 months ago)
Another visualisation for you (for part of the situation). Imagine you want to make an endless toblerone pack out of a flat, rectangular piece of card. You spilt the card into three long rectangles and obvs, the normals are all parallel. When you introduce the two folds, you will fold the card in one plane only, there is no opportunity for the normals to fall out of the vertical plane they were in at the start. Hence, the stay in the same plane.
Hope that wording is clear enough.
If you consider making a sheaf, again, you can think of a series of rectangles that you rotate through different angles and then join each rectangle to one side or other of a line. Whatever you start of with only has to be rotate in one plane and so all the normals remain in the one plane.
Hope that wording is clear enough.
If you consider making a sheaf, again, you can think of a series of rectangles that you rotate through different angles and then join each rectangle to one side or other of a line. Whatever you start of with only has to be rotate in one plane and so all the normals remain in the one plane.
(edited 9 months ago)
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