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# Chapter 11: Vectors and the Geometry of Space

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• 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.
• 11.1E: Exercises for Vectors in the Plane
• 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.
• 11.2E: Exercises for Vectors in Space
• 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.
• 11.3E: Exercises for The Dot Product
• 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.
• 11.4E: Exercises for The Cross Product
• 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.
• 11.5E: Exercises for Equations of Lines and Planes in Space