Uniform acceleration in special relativity

An exercise in Schutz's general relativity book states: "A body is said to be uniformly accelerated if its acceleration four vector $\vec{a}$ has constant spatial direction and magnitude, say $\vec{a}\cdot \vec{a} = \alpha^2$ " . Unfortunately, I don't understand what's meant by "constant spatial direction". I can find an explicit solution if, in a given referential frame, the speed is given by: $\frac{d \vec{x}}{dt} = (1,v(t)\vec{k})$ where $v$ is a function and $\vec{k} \in \mathbb{R}^3$ is a spatial direction. But this is a (very restrictive) condition on speed, not acceleration. Is there a way to generalize this and intrisically define the term "constant spatial direction"? My apologies since this a perhaps more of a math question...