A different vector product I am taking linear algebra, and have learned about the vector dot product and cross product.  Is there a vector product defined by :
    $(a_1, a_2, \dots ,a_n)\times (b_1, b_2, \dots,b_n) = (a_1b_1, a_2b_2, \dots ,a_nb_n)$ ?
If so, what it it called?

This is called the Hadamard product. See http://en.m.wikipedia.org/wiki/Hadamard_product_(matrices)
According to this article, it is also referred to as the Schur product or the entrywise product.
