Problem about Ricci flow

Substituting $Ric=Kg$ into the Ricci flow equation we get $\dot g = -2Kg$, where $\dot g$ is the time derivative of $g$. Since $K$ is a scalar, this equation simply means that every component of $g$ satisfies the same equation (considering $g$ as a matrix): $\dot g_{ik}=Kg_{ik}$, where $i,k=1,2$. Hence, without loss of generality assuming that $g$ is a diagonal matrix, if $K<0$ then $\dot g_{ii}>0$, which means that the length of a vector, say $v$, will grow for a short time at least (more precisely, the time derivative of the length will be positive).