In methane you have four hydrogen atoms bonded to a carbon atom. The four hydrogen atoms form the corner of a tetrahedron. What is the angle between different hydrogen atoms?
In the figure below you see that the atoms of hydrogen can be
put at corners of a cube of width 2. The carbon atom goes in
the middle at coordinate (1,1,1). So we want to determine the
angle 
.
Click here to see full figure
We consider just two of these red bonds in the figure on
the right, and calculate what displacement vectors
these correspond to. They are 
and
.
To calculate the angle between these two vectors we just use the definition of dot product:
But we also know how to calculate the dot product
by multiplying out the components. So 
.
Putting these two things together we have 
,
or 
degrees.