Vector calculus involves representing curves and surfaces in space using parametric equations. Parametric to vector form converts the parameterization to a vector equation, defining the position of a point on the curve or surface as a vector function of a parameter. This transformation enables the analysis of geometric properties, such as velocity, acceleration, and curvature, by calculating derivatives and integrals of the vector function. It plays a crucial role in engineering, computer graphics, motion physics, and optimization, providing insights into the behavior and interactions of objects in our world.