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Find the angle between two planes in space. The only givens are the angles that each plane makes with a line through them both.

JLA
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  • I tried to reduce the problem to the simplest level mathematically, maybe I left out an important bit of info. So what I've actually got is an accelerometer that I've laid on a table on all of its faces and recorded the acceleration. All of them showed almost 9.8 along one axis and very small numbers along the other two. Then the problem was to find the angle between the faces of the accelerometer. I tried defining an arbitrary unit vector normal to the plane of one face and then finding the angle btw it norm and face via dot product, but the choice of normal was affecting the ans. – JLA Mar 12 '13 at 15:33
  • I'm assuming the resultant vectors will all be parallel, even, if the sensing part of the accelerometer is not in the exact middle, because the Earth's center is so far away. – JLA Mar 12 '13 at 15:37

1 Answers1

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This is surely insufficient information. Let $\vec{v}$ be the vector giving us the direction of that auxiliary line. Specifying the angle between that line and one of the planes constrains $\vec{v}$ to lie on a conical surface (centered at the origin) that has the normal of the said plane as a symmetry-axis. So specifying the two angles gives us two cones. Unless the cones are particularly pointy, they will intersect. Therefore we can select the two angles to be anything we want in an interval $(0,\varepsilon)$, where the exact value of $\varepsilon$ depends on the relative directions of the two planes. (The angle between a line and a plane is complementary to the the angle between the line and the normal of the plane, so if the former is close to zero, the cones are very wide, almost planes, and thus two such surfaces will intersect).

Jyrki Lahtonen
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