VECTORS AND FOUNDATIONS UCB Mathematics（向量和基金会的联合数学）.pdf

Chapter 4 VECTORS AND FOUNDATIONS 1 Algebraic operations with vectors 119. Definition of vectors. In physics, some quantities (e.g. distances, volumes, temperatures, or masses) are completely charac- terized by their magnitudes, expressed with respect to a chosen unit by real numbers. These quantities are called scalars. Some others (e.g. velocities, accelerations, or forces) cannot be characterized only by their magnitudes, because they may differ also by their directions. Such quantities are called vectors. B u D A u C Figure 121 To represent a vector quantity geometrically, we draw an arrow connecting two points in space, e.g. A with B (Figure 121). We call it a directed segment with the tail A and head B , and indicate −−→ this in writing as AB . 107 108 Chapter 4. VECTORS AND FOUNDATIONS The same vector can be represented by different directed seg- −−→ −−→ ments. By definition, two directed segments (AB and CD) rep- resent the same vector if they are obtained from each other by translation (§72). In other words, the directed segments must have the same length, lie on the same line or two parallel lines, and point toward the same direction (out of two possible ones). When this is −−→ −−→ the case, we write AB = CD and say that the vectors represented −−→ −−→ by these directed segment