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What'southward the best way to depict two planes intersecting at an bending that isn't $\pi /two$?

If I make them both vertical and vary the angle between them, the diagram e'er looks as though our viewpoint has changed but the planes are however intersecting at $\pi /2$.

I can't quite piece of work out how to describe 1 or both of them non-vertical in such a way as to make the angle between them appear to be obviously not a correct angle.

Thank you for whatsoever assist with this!

asked April 17, 2012 at nine:35

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5

  • $\begingroup$ One good method is to accept the dot product of their unit normal vectors, and take the arc cosine of that to get the bending between the planes, as in this related question. In detail, the planes are perpendicular iff the dot product of their normal vectors is zero. Besides, the airplane $ax+by+cz=d$ has normal vector $(a,b,c)$. $\endgroup$

    Apr 17, 2012 at ix:38

  • $\begingroup$ If yous were to look at the intersection from the line of intersection, the planes would conspicuously announced to intersect at an bending other than 90 degrees(provided they don't intersect at xc degrees). $\endgroup$

    Apr 17, 2012 at 9:42

  • $\begingroup$ @bgins - apologies for causing confusion - I meant to enquire well-nigh drawing them, not 'showing' non-orthogonality in the mathematical sense. I've now amended the title and question to make this clearer $\endgroup$

    April 17, 2012 at ix:55

  • $\begingroup$ @BenEysenbach - unfortunately I tin't do that, because I need to prove ii distinct points on the line of intersection $\endgroup$

    Apr 17, 2012 at 9:56

  • $\begingroup$ One way would exist to take an astute triangle and extend the larger sides into planes, sometthing like here. Another would exist to draw several intersecting radial lines and extend them all to planes, peradventure using color, something like hither or here. Lastly, you lot might try drawing a parallelopiped (similar here) and refer to the planes of the faces. $\endgroup$

    Apr 17, 2012 at 10:07

one Answer i

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Here's my attempt, along with a few ideas I've applied in my drawings for multivariable calculus.

  • It helps to kickoff with ane of the planes completely horizontal, or at least close to horizontal-- then everything else you lot draw will exist judged in relation to that.
  • Probably the most important thing is to use perspective. Parallel lines, like contrary 'edges' of a plane, should non be drawn every bit parallel. In an image correctly drawn in perspective, lines that meet at a common, far-off signal will appear to be parallel. Discover the iii lines in my horizontal plane that will run into far away to the upper-left of the drawing. This forces you to interpret the lower-correct edge equally the near edge of the plane. I sometimes use thicker or darker lines to indicate the about border, but perspective is a much more dominant force. Information technology helps you translate the drawing even if information technology's non perfectly done, every bit often happens when I'grand cartoon on the board.
  • You can 'cheat' past copying real objects. I started this drawing by studying my laptop from an odd angle, and reproducing the planes defined past the keyboard and screen.
  • Any actress lines showing the 'filigree lines' of each plane will assistance. Whenever I talk about normal vectors, I ever describe a little plus sign on the plane to ballast them.
  • The intersection line of the two planes can be totally arbitrary- notice that mine appears parallel with edges of the horizontal plane, only not quite parallel with any edges of my skew aeroplane. You can experiment with different angles and lines of intersection; many of them will yield dainty drawings.

not normal planes

answered Apr 17, 2012 at 14:46

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