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Angles
Line-Line Angles

The angle θ between two lines is the angle between two direction vectors of the lines.
Take line r with direction vector u and line s with direction vector v, then

 The angle between the two lines is defined by:

Calculate the angle between the following lines:

$r\;\equiv\;\frac{x-2}{2}=\frac{y+1}{1}=\frac{z}{1}\;$ and $s\;\equiv\;\frac{x+1}{-1}=\frac{y}{2}=\frac{z}{1}\;$

Solution:

u=(2,1,1) and v=(-1,2,1)

Therefore

θ=80.41º

To find the angle between the line

 2x+3y-z+1=0 x-y+2z+2=0

and the line

 3x-y+z-3=0 2x+y-3z+1=0

Solution:

u=
 i j k 2 3 -1 1 -1 2
=5i-5j-5k $\Rightarrow$ u=(5,-5,-5)

v=
 i j k 3 -1 1 2 1 -3
=2i+11j+5k $\Rightarrow$ v=(2,11,5)

θ=48.70º