  In this paper we study the cardinality of the dot product set generated by
two subsets of vector spaces over finite fields. We notice that the results on
the dot product problems for one set can be simply extended to two sets. Let E
and F be subsets of the d-dimensional vector space F_q^d over a finite field
F_q with q elements. As a new result, we prove that if E and F are subsets of
the paraboloid and |E||F|\geq C q^d for some large C>1, then |\Pi(E,F)|\geq c q
for some 0<c<1. In particular, we find a connection between the size of the dot
product set and the number of lines through both the origin and a nonzero point
in the given set E. As an application of this observation, we obtain more
sharpened results on the generalized dot product set problems. The discrete
Fourier analysis and geometrical observation play a crucial role in proving our
results.
