Answer
$-pq$
Work Step by Step
The dot product of two vectors $\mathbf a= \langle a_1,a_2,a_3 \rangle$ and $\mathbf b= \langle b_1,b_2,b_3 \rangle$ is $a_1b_1+a_2b_2+a_3b_3$.
So for $\mathbf a= \langle p,-p,2p \rangle$ and $\mathbf b= \langle 2q,q,-q \rangle$, $$\mathbf a \cdot \mathbf b =p(2q)+(-p)(q)+2p(-q)=2pq-pq-2pq=-pq.$$