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Chapter 11: Vectors and the Geometry of Space

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Jan 14, 2025
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- Section 10.5: Chapter 10 Review Exercises
- Section 11.0: Prelude to Vectors in Space

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Section 11.0: Prelude to Vectors in Space
Section 11.1: Vectors in the Plane
Some quantities, such as or force, are defined in terms of both size (also called magnitude) and direction. A quantity that has magnitude and direction is called a vector.
- Section 11.1E: Exercises for Vectors in the Plane
Section 11.2: Vectors in Space
Vectors are useful tools for solving two-dimensional problems. Life, however, happens in three dimensions. To expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space.
- Section 11.2E: Exercises for Vectors in Space
Section 11.3: The Dot Product
The dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. The dot product can also help us measure the angle formed by a pair of vectors and the position of a vector relative to the coordinate axes. It even provides a simple test to determine whether two vectors meet at a right angle.
- Section 11.3E: Exercises for The Dot Product
Section 11.4: The Cross Product
In this section, we develop an operation called the cross product, which allows us to find a vector orthogonal to two given vectors. Calculating torque is an important application of cross products, and we examine torque in more detail later in the section.
- Section 11.4E: Exercises for The Cross Product
Section 11.5: Equations of Lines and Planes in Space
To write an equation for a line, we must know two points on the line, or we must know the direction of the line and at least one point through which the line passes. In two dimensions, we use the concept of slope to describe the orientation, or direction, of a line. In three dimensions, we describe the direction of a line using a vector parallel to the line. In this section, we examine how to use equations to describe lines and planes in space.
- Section 11.5E: Exercises for Equations of Lines and Planes in Space
Section 11.6: Quadric Surfaces
We have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes and spheres, which are examples of three-dimensional figures called surfaces, to explore a variety of other surfaces that can be graphed in a three-dimensional coordinate system.
Section 11.7: Cylindrical and Spherical Coordinates
The Cartesian coordinate system provides a straightforward way to describe the location of points in space. Some surfaces, however, can be difficult to model with equations based on the Cartesian system. As the name suggests, cylindrical coordinates are useful for dealing with problems involving cylinders. Similarly, spherical coordinates are useful for dealing with problems involving spheres.
Section 11.8: Chapter 11 Review Exercises

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

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