Week 1: Three-Dimensional Coordinate Systems, Vectors, The Dot Product.

Thomas Calculus, 11th edition (page 848 - 872)

Sec 12.1 Three Dimensional Coordinate System

To apply calculus in many real-world situations and in higher mathematics, we need a mathematical description of three-dimensional space. In these sections, we introduce three-dimensional coordinate systems and vectors. Building on what we already know about coordinates in the xy-plane, we establish coordinates in space by adding a third axis that measures distance above and below the xy-plane. Vectors are used to study the analytic geometry of space, where they give simple ways to describe lines, planes, surfaces, and curves in space.

Sec 12.2 Vectors

Some of the things we measure are determined simply by their magnitudes. To record mass, length, or time, for example, we need only write down a number and name an appropriate unit of measure. We need more information to describe a force, displacement, or velocity. To describe a force, we need to record the direction in which it acts as well as how large it is. To describe a body’s displacement, we have to say in what direction it moved as well as how far. To describe a body’s velocity, we have to know where the body is headed as well as how fast it is going.

Sec 12.3 The Dot Product

In this section, we show how to calculate easily the angle between two vectors directly from their components. A key part of the calculation is an expression called the dot product. Dot products are also called inner or scalar products because the product results in a scalar, not a vector. After investigating the dot product, we apply it to finding the projection of one vector onto another (as displayed in Figure 12.18) and to finding the work done by a constant force acting through a displacement.