Why They're Called Direction Cosines

Date: 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!