What is the purpose of the dot product as opposed to the purpose of the cross product?
the result of a dot product of two vectors is a scalar ...
work, a scalar quantity of energy, is the dot product of force and displacement vectors
$\displaystyle W = \vec{F} \cdot \vec{\Delta x}$
the result of a cross product of two vectors is another vector ...
torque, a vector quantity, is the cross product of radius and force vectors
$\displaystyle \vec{\tau} = \vec{r} \times \vec{F}$
try a google search ... you'll find many more examples and applications.