Equation for Angle Formed by Two Vectors

Date: 05/31/2000 at 03:24:07
From: Michelle
Subject: 3D vectors

If I have two vectors, e.g. OP and OQ, to find the angle POQ I know I 
have to use the formula cos(theta) = a.b / ab (the denominator 
represents the scalar product.)

However, before I can use this I need to obtain values of a and 
b. Do I need to find the vector equation of OP and OQ or PO and OQ?

Date: 05/31/2000 at 06:04:15
From: Doctor Pat
Subject: Re: 3D vectors

Michelle,

To find the angle at O, you need two vectors each with a foot or base 
at O, so you need the vectors OP and OQ. You can also use PO and QO, 
which are the negatives of the first two, and get the same angle, but 
if you are having trouble with these at first, you might want to make 
a rule of thumb to use the vectors FROM THE VERTEX. If you mix these 
up you will get the supplement of the angle you seek...

            A    B 
             \  /
              \/
              C
              /\
             /  \
            D    E

For angle ACB you can use vectors CA and CB. But from the picture you 
can see that BC and AC are the same directions as CE and DE so you 
would get the vertical angle DCE, which would be the same measure.

If you use CB and AC, that is the same as CB and CE, which gives angle 
BCE, the supplement of ACB.  

Hope that helps you see how the dot product works. Good luck.

- Doctor Pat, The Math Forum
  http://mathforum.org/dr.math/