Why They're Called Direction CosinesDate: 09/11/2003 at 21:32:05 From: Kristen Subject: Vector Angles I would like to know how to find the angles between a 3D vector and the 3 coordinate axes, given the components of the vector. Date: 09/12/2003 at 08:36:36 From: Doctor Ian Subject: Re: Vector Angles Hi Kristen, If you have two vectors, A and B, and you want to find the angle between them, one way is to use the dot product: dot(A,B) = |A||B|cos(theta) Does that look familiar? To find the angle between a vector and a particular axis, you can just make B a unit vector. For example, if A is (a,b,c), then to find the angle with the x-axis, _______________ a*1 + b*0 + c*0 = \|a^2 + b^2 + c^2 cos(theta) a -------------------- = cos(theta) _______________ \|a^2 + b^2 + c^2 Note that if you make A a unit vector (which you can do by dividing all the components by the magnitude of A), you end up with a b c ( ---, ---, --- ) = ( cos(theta ), cos(theta ), cos(theta ) ) |A| |A| |A| x y z For this reason, the components of a unit vector are often called the 'direction cosines' of the vector. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 09/12/2003 at 11:07:54 From: Kristen Subject: Thank you (Vector Angles) Thank you for your help. That will make my life SO much easier! |
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