The use of vectors not only simplifies treatments of differential geometry, mechanics, hydrodynamics, and electrodynamics, but also makes mathematical and physical concepts more tangible and easy to grasp. This text for undergraduates was designed as a short introductory course to give students the tools of vector algebra and calculus, as well as a brief glimpse into these subjects' manifold applications. The applications are developed to the extent that the uses of the potential function, both scalar and vector, are fully illustrated. Moreover, the basic postulates of vector analysis are brought to the foreground, placing their logical structure in sharp relief. Because the concept of a vector has been greatly generalized in geometry and mathematical physics, this text concludes with a brief introduction to abstract vector spaces, together with the ideas of linear dependence, basis, and dimension. The exposition of these abstract concepts is kept simple and clear. Numerous figures appear throughout the text.