Likewise, "vector" is an interface, supporting vector addition, scalar multiplication, and (crucially in this context) inner, wedge and geometric products. Many data structures can implement this interface (e.g. a tuple of scalars, as coefficient of a fixed orthonormal basis; a scalar length paired with angular spreads projected in fixed perpendicular planes; etc.)
> Using the terms "scalar with a direction" to describe a vector makes no sense if you come from that starting point.
Now I'm confused: what is a vector, if not a "scalar with a direction" (or indeed "an oriented scalar value", as the parent quoted)?
> 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.
I disagree; "scalar value" is an interface, supporting addition, multiplication, negation, etc. (e.g. https://hackage.haskell.org/package/base-4.16.1.0/docs/Prelu... https://smlfamily.github.io/Basis/real.html ). Many data structures can implement this interface (Float, Double, Rational, Fixed, Nat, Int, etc.).
Likewise, "vector" is an interface, supporting vector addition, scalar multiplication, and (crucially in this context) inner, wedge and geometric products. Many data structures can implement this interface (e.g. a tuple of scalars, as coefficient of a fixed orthonormal basis; a scalar length paired with angular spreads projected in fixed perpendicular planes; etc.)