x, y, and z are just shorthand for orthonormal basis vectors. What you’ve described isn’t “traveling in all three dimensions” simultaneously, it’s traveling along on of the dimensions with a different basis.
Undoubtedly there must be something I’m missing here—I’ve taken physics courses but clearly I’m no expert. :)
That is, any vector is going to be inherently one dimensional, regardless of its coordinates. Dimension is a property of a set of vectors, dependent upon how many are linearly independent of the rest of the vectors in the set. What I described doesn’t imply that there are infinite dimensions.
The vector [0,0,1] is traveling through three dimensions just the same as [1,1,1].