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In this paper we confirm a conjecture of F\"uredi, Jiang, and Seiver, anddetermine an exact formula for the Tur\'an number $ex_3(n; P_3^3)$ of the3-uniform linear path $P^3_3$ of length 3, valid for all $n$. It coincides withthe analogous formula for the 3-uniform triangle $C^3_3$, obtained earlier byFrankl and F\"uredi for $n\ge 75$ and Cs\'ak\'any and Kahn for all $n$. In viewof this coincidence, we also determine a `conditional' Tur\'an number, definedas the maximum number of edges in a $P^3_3$-free 3-uniform hypergraph on $n$vertices which is \emph{not} $C^3_3$-free.

Tur\'an numbers for 3-uniform linear paths of length 3