The meaning of ds and differential forms

Question

On page 162 of Anderson's, Fundamentals of Aerodynamics the equation of a streamline of a flow is developed. In this section, $\boldsymbol {ds}$ is said to be "a directed element of the streamline". Then it is stated that in cartesian coordinates, $$\boldsymbol {ds} = dx \hat{\imath} + dy \hat{\jmath} + dz \hat{k}$$ In this answer to What is the rigorous definition of dy and dx?, $dx$, $dy$, and $dz$ are said to be linear functionals which form the standard basis of $(\mathbb R^3)'$. If this is the case then are there any meaningful definitions for the multiplication of $dx$ and $\hat{\imath}$?