In this chapter, the basic and advanced problems of cartesian, cylindrical, and spherical coordinate systems are studied. The subjects include unit vector; dot product (inner product); cross product (vector product); differential length vectors, differential area vectors, and differential volume in cartesian, cylindrical, and spherical coordinate systems; conversions between the coordinate systems; position vector; and distance vector. Herein, different types of problems and exercises are presented that are categorized as follows.