In this exposition, Emil Artin—one of the foremost mathematicians of the 20th century and a gifted expositor—provides a systematic exploration of fundamental concepts in analytic geometry, guiding the reader from the basic notions of points and their numerical representation in space to the more sophisticated definitions of vectors, distances, and angles. Beginning with a reflective analysis of how the abstract ideal of a point relates to measurable physical phenomena, the discussion naturally progresses toward precise mathematical formulations. Through careful derivations—illustrated with intuitive diagrams—the reader revisits essential geometric results such as the theorem of Pythagoras, and further extends these principles into higher-dimensional spaces. The text also introduces vectors rigorously, clarifying their operations, properties, and interpretations in terms of length and direction. It culminates with applications to geometric entities such as lines, planes, and their intersections, providing methods to calculate distances and angles using vectors, scalar products, and vector products. This approach not only emphasizes mathematical clarity but also equips the reader with versatile analytical tools applicable across dimensions.