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Mathematics LibreTexts

10: Vectors

This chapter introduces a new mathematical object, the vector. We will see that vectors provide a powerful language for describing quantities that have magnitude and direction aspects. A simple example of such a quantity is force: when applying a force, one is generally interested in how much force is applied (i.e., the magnitude of the force) and the direction in which the force was applied. Vectors will play an important role in many of the subsequent chapters in this text. This chapter begins with moving our mathematics out of the plane and into "space.'' That is, we begin to think mathematically not only in two dimensions, but in three. With this foundation, we can explore vectors both in the plane and in space. 

  • 10.1: Introduction to Cartesian Coordinates in Space
    In this section we introduce Cartesian coordinates in space and explore basic surfaces. This will lay a foundation for much of what we do in the remainder of the text. Each point P in space can be represented with an ordered triple, P=(a,b,c), where a,b and c represent the relative position of PP along the x, y and z -axes, respectively. Each axis is perpendicular to the other two.
  • 10.2: An Introduction to Vectors
    Many quantities we think about daily can be described by a single number: temperature, speed, cost, weight and height. There are also many other concepts we encounter daily that cannot be described with just one number. For instance, a weather forecaster often describes wind with its speed and its direction. When applying a force, we are concerned with both the magnitude and direction of that force. In both of these examples, direction is important.
  • 10.3: The Dot Product
    The previous section introduced vectors and described how to add them together and how to multiply them by scalars. This section introduces a multiplication on vectors called the dot product.
  • 10.4: The Cross Product
    "Orthogonality'' is immensely important. Given two non--parallel, nonzero vectors u and v in space, it is very useful to find a vector w that is perpendicular to both u and v. There is a operation, called the cross product, that creates such a vector. This section defines the cross product, then explores its properties and applications.
  • 10.5: Lines
    To find the equation of a line in the x-y plane, we need two pieces of information: a point and the slope. The slope conveys direction information. As vertical lines have an undefined slope, the following statement is more accurate: "To define a line, one needs a point on the line and the direction of the line."
  • 10.6: Planes
    Any flat surface, such as a wall, table top or stiff piece of cardboard can be thought of as representing part of a plane.


  • Gregory Hartman (Virginia Military Institute). Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License.