I fail to see how that makes any sense. OP clearly referred to the unidimensional nature of a scalar in contrast to the n-dimensional nature of a vector, or the n*m-dimensional nature of a matrix. It makes no sense to try to go off on a tangent regarding, say, the resolution of a scalar representation. Perhaps the concept of a ray gets halfway there. Perhaps the concept of a line at infinity.
The concept of a scalar is familiar, as is the concept of a vector. Using the terms "scalar with a direction" to describe a vector makes no sense if you come from that starting point. Perhaps magnitude+direction vector rings a bell, because that's also a basic description of a vector.
Perhaps the author made a mistake, or does not have a good math background, or is filing in his knowledge gaps by coming up with definitions . Or perhaps he's actually referring to different concepts that the readership is not familiar. Who knows?
> This is tongue-in-cheek; but from a software perspective, all those representations have the same "API" (arithmetic).
This assertion is quite wrong. Both scalar and vector, in this context, are data types. Describing a data types as a data type of a data type is meaningless and does not compute.