Math behind Multi-class linear discriminate analysis (LDA)
Given normalized top $k$ eigenvectors $v_1,\ldots,v_k$, you send a point $x$ to the tuple $(\langle x,v_1 \rangle, \ldots, \langle x,v_k \rangle)$.

Alternatively, you put the eigenvectors as rows in a matrix $M$, and you map the column vector $x$ to $Mx$ (this is exactly the same thing as above).

Multiplying vectors is an operation that doesn't have much semantic sense here. (How would you multiply them anyway, and why?)

There are reasons why we apply this particular operation, and they have to do with the goal of LDA. If you want to know more, I suggest picking up a decent textbook that explains LDA not only via formulas and algorithms, but also as the solution to an optimization problem.