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March 22nd, 2010, 01:50 PM  #1 
Newbie Joined: Mar 2010 Posts: 4 Thanks: 0  Solving a triangle from vectors
I have a triangle defined by three vectors. The first vector, r, has a known direction and magnitude. The second vector, f, has a known direction. The third vector, g, has a known magnitude. My goal is to solve for the length of f. What I have tried so far is: 1) Use cross product of f's direction and r to get the angle between them. 2) Use sine law with magnitude of g to find angle between g and r. 3) Subtract the existing angles from 180 to get the angle across from f. 4) Use sine law to calculate length of f. This has not worked, however. Is there another method that would work better? Am I making an incorrect assumption somewhere? Thanks for any help. 
March 22nd, 2010, 02:26 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,931 Thanks: 1125 Math Focus: Elementary mathematics and beyond  Re: Solving a triangle from vectors
Law of cosines, perhaps? If you know two directions, can you get the angle between those two vectors and use that and the magnitude of the third vector to solve for the magnitude of f?

March 22nd, 2010, 03:29 PM  #3  
Newbie Joined: Mar 2010 Posts: 4 Thanks: 0  Re: Solving a triangle from vectors Quote:
 
March 22nd, 2010, 03:33 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,931 Thanks: 1125 Math Focus: Elementary mathematics and beyond  Re: Solving a triangle from vectors
You'll need the angle between r and f. 0 = f^2 2rfcos? + r^2  g^2, where ? is the angle between r and f. 
March 23rd, 2010, 07:29 AM  #5 
Newbie Joined: Mar 2010 Posts: 4 Thanks: 0  Re: Solving a triangle from vectors
Thanks ... that helps a bunch! It's been about six years since I did this stuff in high school and my head doesn't quite go through them as fast as it used to, which is a shame. I need to get this formula into a state where f = something, so I can use code to calculate that value. But I can't remember how to get f on it's own when I have cf + f^2 on one side. Any tips to help me get started?

March 23rd, 2010, 07:56 AM  #6 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,931 Thanks: 1125 Math Focus: Elementary mathematics and beyond  Re: Solving a triangle from vectors
Consider 0 = f^2 2rfcos? + r^2  g^2. It is a quadratic in f. Having values for r, g and cos? allows you to write 0 = af^2 + bf + c, where a = 1, b = 2rcos?, c = r^2  g^2. Quadratic formula: . This gives two possibilities for f. 
March 23rd, 2010, 08:23 AM  #7 
Newbie Joined: Mar 2010 Posts: 4 Thanks: 0  Re: Solving a triangle from vectors
Thanks Greg  your help has been invaluable!

March 23rd, 2010, 05:48 PM  #8  
Global Moderator Joined: Dec 2006 Posts: 20,469 Thanks: 2038  Quote:
 

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