To apply calculus in many real-world situations and in higher mathematics, we need a mathematical description of three-dimensional space. In this chapter 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 study the analytic geometry of space, where they give simple ways to describe lines, planes, surfaces, and curves in space. We use these geometric ideas in the rest of the book to study motion in space and the calculus of functions of several variables, with their many important
applications in science, engineering, economics, and higher mathematics
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. A quantity such as force, displacement, or velocity is called a vector and is represented by a directed line segment The arrow points in the direction of the action and its length gives the magnitude of the action in terms of a suitably chosen unit. For example a force vector points in the direction in which the force acts; its length is a measure of the force’s strength; a velocity vector points in the direction of motion and its length is the speed of the moving object. displays the velocity vector v at a specific location for a particle moving along a path in the plane or in space. This application of vectors is studied in Chapter The arrows we use when we draw vectors are understood to represent the same vector
if they have the same length, are parallel, and point in the same direction regardless of the initial point.
In textbooks, vectors are usually written in lowercase, boldface letters, for example u, v, and w. Sometimes we use uppercase boldface letters, such as F, to denote a force vector. In handwritten form, it is customary to draw small arrows above the letters, for example and We need a way to represent vectors algebraically so that we can be more precise about
the direction of a vector. Let There is one directed line segment equal to whose initial point is the
origin It is the representative of v in standard position and is the vector
we normally use to represent v. We can specify v by writing the coordinates of its terminal point when v is in standard position. If v is a vector in the plane its terminal
In studying lines in the plane, when we needed to describe how a line was tilting, we used the notions of slope and angle of inclination. In space, we want a way to describe how a plane is tilting. We accomplish this by multiplying two vectors in the plane together to get
a third vector perpendicular to the plane. The direction of this third vector tells us the “inclination” of the plane. The product we use to multiply the vectors together is the vector or cross product, the second of the two vector multiplication methods we study in calculus.
Cross products are widely used to describe the effects of forces in studies of electricity, magnetism, fluid flows, and orbital mechanics. This section presents the mathematical properties that account for the use of cross products in these fields