Light Beam Reflection
Date: 01/31/2002 at 07:02:08 From: Teri Lurie Subject: Algebra Please can you help me with this question? Four mirrors form a rectangle 3 m by 2 m. A light beam is shone from A at 45 degrees. Which corner does the beam strike first? Teri
Date: 01/31/2002 at 12:17:13 From: Doctor Peterson Subject: Re: Algebra Hi, Teri. You can either do this in a very simple way, by just drawing what happens, or in a way that demands more insight but makes the problem a lot more interesting. First, you can just draw it out: B C +---+---o---+ | | / | | +---/---+---+ | / | | | o---+---+---+ A D As I've shown, the beam will first hit the wall 2/3 of the way along side BC, at coordinates (2,2) if A is the origin. Then it will reflect at the same 45 degree angle back down toward side CD. Draw where it will hit that: B C +---+---o---+ | | / | \ | +---/---+---o | / | | | o---+---+---+ A D (It will be very useful to realize that a 45 degree angle will always take the beam one unit in the y direction for each unit it goes in the x direction.) Keep on like this, and you will soon find the answer. Now for some deeper insight: Instead of thinking of the beam reflecting, you can think of the reflections of the "pool table" itself (that's how I think of it) in the mirrors: A---+---+---D---+---+---A | | | | | | | +---+---+---o---+---+---+ | | | / | | | | B---+---o---C---+---+---B | | / | | | | | +---/---+---+---+---+---+ | / | | | | | | A---+---+---D---+---+---A If you were standing on the "table," you would see many copies of the table (and yourself) over and over. And the path of a light beam would seem to be a straight line cutting through many of these reflected copies. I've shown its path from A to side BC and on to a reflected copy of side CD, matching what I showed before. Keep going, and it will eventually hit a reflection of a corner, and that will be your answer. Now if you think about this a bit, you will see that the corner it hits will be at coordinates (x,y) where x and y are equal (so that it is on the path of the light beam), but x is a multiple of 3 and y is a multiple of 2 (so that it is a reflection of a corner). Can you see how to use this fact to solve the problem without having to follow the beam all the way? That would be very useful for solving the problem more generally, for different sized tables and different directions. You can find some related problems in our archive at Bouncing Cue Ball http://mathforum.org/dr.math/problems/nichole10.29.96.html Pool Table Algebra http://mathforum.org/dr.math/problems/roland10.21.98.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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