This chapter describes the motion of particles in two and three dimensions using Cartesian coordinates. Describing motion in two and three dimensions requires the use of vectors. After reviewing the basic properties of vectors and introducing the dot and cross products, the chapter introduces vector derivatives. Vector derivatives are necessary in order to compute the velocity and acceleration vectors. The chapter then describes polar, cylindrical, and spherical coordinates, and demonstrates how to describe a particle’s position, velocity, and acceleration in those coordinate systems. It discusses how to describe a particle’s motion in coordinate systems other than Cartesian coordinates. The new coordinate systems often exploit symmetries in the problem, making the non-Cartesian coordinate systems a more natural means of describing the motion of the particle. Finally, the chapter concludes with special vector derivatives commonly used in all fields of physics, such as the gradient, the divergence, and the curl.