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I'm solving a system defined by two rays. The rays originate from two points {x,y,z} and {x,y,z-h}. They go in directions (unit vectors) {u1,v1,w1} and {u2,v2,w2}. The rays are connected by a line segment of length l. The endpoints of the line segment are t units down the length of each ray. The variable to solve for is t.

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This solution is all well and good, but I notice in the output I have collections of terms like u1^2 ... + v1^2 ... + w1^2. I know that {u1,v1,w1} and {u2,v2,w2} are unit vectors, so these could be substituted with a 1, and maybe even other simplifications could occur. I modified the solve like so:

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This looks like a step in the right direction. (I tried Norm[{u1,v1,w1}]==1 but it had no effect.) So, I tried constraining {u2,v2,w2} to a unit vector.

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Sadly, it blows up to a size even larger than the first output. Doing this last step manually could be "an exercise left up to me," but I'm curious if there's some way I could approach this that yields better results. I thought maybe there was some way to declare a vector as a unit vector, and came across How can I use a unit vector notation found in physic texts? but I wasn't able to glean anything useful from it. How can I let Mathematica know that a vector is a unit vector so that it can use it as a hint in simplifying output?

Brent
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    Welcome to Mathematica.SE! I suggest the following:
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     – Dunlop Mar 22 '20 at 05:16
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    Solve by itself does not do everything it can to Simplify a result. That makes Solve faster and you can use Simplify only if need be. This FullSimplify[(-h(2w1-2w2)-Sqrt[h^2(2w1-2w2)^2-4(h^2-l^2)(u1^2-2u1*u2+u2^2+v1^2-2v1*v2+v2^2+w1^2-2w1*w2+w2^2)])/(2(u1^2-2u1*u2+u2^2+v1^2-2v1*v2+v2^2+w1^2-2w1*w2+w2^2)), Norm[{u1,v1,w1}]==1&& Norm[{u2,v2,w2}]==1] gives a substantially smaller result, but only because I have included the second argument to FullSimplify telling it about two of your assumptions, otherwise Mathematica would never know. With more assumptions an even smaller result? – Bill Mar 22 '20 at 05:33
  • Try doing this Replace[Solve[{EuclideanDistance[r1, r2] == l}, t], {u1^2 -> 1 - v1^2 - w1^2, u2^2 -> 1 - v2^2 - w2^2}, {6, 10}] // Simplify . The idea is that you replace the solution at different "levels" inside the equation. To understand this more check out the help for Replace[] – Dunlop Mar 22 '20 at 05:48

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