This textbook is written to develop thorough understanding the mastery behind the foundations of geometry that needed to solve geometric problems using vectors in three dimensional Euclidean space at a useful proficiency level, it introduces a comprehensive study on three dimensional Euclidean spaces'' objects, namely; points, line segments, vectors, straight lines, planes, spheres, cylinders, quadratic surfaces in general analytically, and includes different comparisons between such studied objects among themselves, it focuses on the orthogonal projections of a point and directed line segment on line, on plane, and on sphere, and it is recommended world widely for the second level universities'' students, its contents are written in a systematic, streaming, understandable and attractive way for student''s interest, supported with adequate number of examples, exercises, problem solving and applications to meet the quality standard. Moreover; it is a prerequisite for some other fields o
