Rotation Matrix Using Trig FunctionsDate: 12/15/2005 at 21:36:46 From: Jacob Subject: Rotation Matrix using trigonometry My question was given to me to think about by my geometry teacher: Why is: |cos(theta) -sin(theta)| |sin(theta) cos(theta)| where theta is the angle of rotation, the rotation matrix? I have tried graphing each part of the matrix in a polar graph, and think I may be onto something, but don't know what to do with it (I got a circle rotated 90 degrees twice counter-clockwise around the origin, with diameter 1). I have also tried treating the matrix as a vertex matrix and finding the equation of the line, but no luck. A little help would be greatly appreciated. Date: 12/16/2005 at 16:36:04 From: Doctor George Subject: Re: Rotation Matrix using trigonometry Hi Jacob, Thanks for writing to Doctor Math. Compute the point (x',y') where |x'| = |cos(theta) -sin(theta)| |x| |y'| |sin(theta) cos(theta)| |y| Now compare the segment from (0,0) to (x,y) with the segment from (0,0) to (x',y'). Investigate the lengths of the segments and the angle between them. Write again if you need more help. - Doctor George, The Math Forum http://mathforum.org/dr.math/ Date: 12/17/2005 at 12:04:23 From: Jacob Subject: Thank you (Rotation Matrix using trigonometry) Thanks a lot! I had tried going down that road, but I got lost, if you know what I mean. You've shown me the "light". Thank you. I highly appreciate the work you guys (and gals) do at the Math Forum. Keep up the good work! Date: 11/26/2007 at 20:13:18 From: Sakura Subject: Re:Rotations of Matrix Using Trig Functions in Simpler Terms I read your question about rotations of matrices using trig functions. (Note: this has only to do with rotating in origin). However....... For some reason, I still do not understand why |cos(beta) -sin(beta)| |sin(beta) cos(beta)| is the one used for rotation. Maybe I am not being very clear. I understand how this matrix was acquired, but I do not understand the application, reason, and origin of why a trig function has to be involved for rotation. This is really hard to explain not having a graphing paper included. But, I know that when one rotates something, the original (x,y) is switched to (-y,x). And using some inverse matrix equations, one gets the above. But just why does there have to be trig functions? Date: 11/28/2007 at 08:52:15 From: Doctor George Subject: Re:Rotations of Matrix Using Trig Functions in Simpler Terms Hi Sakura, Thanks for writing to Doctor Math. Rotation matrices are worth the effort you are putting into understanding them. You seem to be starting on them at a younger age than most, so I'm not sure what you have studied so far. Hopefully you have learned about vectors. Each row of the matrix represents a vector. cos(beta) i - sin(beta) j sin(beta) i + cos(beta) j Notice that these vectors have unit length and are rotated by an angle beta from the coordinate axes. They are also perpendicular to each other. When you multiply a vector x times the matrix you are actually taking the dot product of x and each of the row vectors. The result is the component of x in the directions of the row vectors. So the trig functions give us a clean way to produce vectors that have been rotated by a specified angle. Does that make sense? Write again if you need more help. - Doctor George, The Math Forum http://mathforum.org/dr.math/ |
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