maximizing inner product
For three-dimensional vectors, construct the three-dimensional convex hull of the vectors in $L'$ in time $O(n\log n)$. The maximizer for a vector $v$ in $L$ is the point of the convex hull that is most extreme in the direction of $v$, and extreme-vertex queries may be answered in logarithmic time by using a Dobkin-Kirkpatrick hierarchy for the convex hull. For this part see e.g. O'Rourke's Computational Geometry in C, pp. 272ff. So the 3d problem can be solved in total time $O(n\log n)$.

For 4-dimensional vectors, the same thing would work, but you don't want to construct the convex hull because it's too expensive. Instead, it is possible to process a set of $n$ points in time and space $\tilde O(m)$ (for any $m$ between $n$ and $n^2$) so that you can answer extreme-point queries in time $\tilde O(n/\sqrt{m})$; see Corollary 8(iii) of Agarwal and Erickson's range query survey. Choosing $m=n^{4/3}$ gives total time $\tilde O(n^{4/3})$ to set up a data structure for $L'$ and query all vectors in $L$.