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Unformatted text preview: Chapter 11: Three Dimensional Space; Vectors Summary: This chapter introduces three dimensional spaces and the concept of a vector. Much of the material here is focused upon describing vectors, how they interact with each other and their significance in both two and three dimensional spaces. Dot products and cross products of vectors are two of the most signif- icant ideas in the chapter. These two features of vectors allow many geometric relationships to be built about planes, curves and later, vector-valued functions. Using dot products, planes are defined in terms of normal vectors and both dot and cross products give information about the angle between two vectors. Quadric sur- faces are also discussed in the chapter. These are extensions of conic sections from two dimensions to surfaces in three dimensions. Finally, cylindrical and spherical coordinate systems are introduced as alternative ways of describing three dimen- sional spaces. Both cylindrical and spherical coordinates are similar in nature to polar coordinates that were introduced in two dimensional space as an alternative to rectangular coordinates. OBJECTIVES: After reading and working through this chapter you should be able to do the following: 1. Identify different surfaces in three dimensions such as → spheres (§11 . 1), → cylindrical surfaces (§11 . 1), → planes (§11 . 6), → quadric surfaces (§11 . 7). 2. Find a vector based upon two points (§11 . 1). 3. Calculate the norm of a vector, a scalar product and the sum of two vectors (§11 . 2). 4. Normalize a vector (§11 . 2). 5. Find the angle between two vectors (§11 . 3 , 11 . 4). 6. Calculate the dot product of two vectors (§11 . 3). 7. Calculate the cross product of two vectors (§11 . 4). 227 228 8. Calculate the scalar triple product of three vectors (§11 . 4). 9. Find the parametric equation of a line in the direction of a vector (§11 . 5). 10. Find the equation of a plane (§11 . 6). 11. Calculate various distances between planes, points and lines (§11 . 6). 12. Identify and sketch quadric surfaces (§11 . 7). 13. Convert between the rectangular, cylindrical and spherical coordinate sys- tems (§11 . 8). 11.1 Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces PURPOSE: To extend rectangular coordinates to three dimen- sions and to introduce the equation of a sphere and some cylin- drical surfaces that are extensions of curves in a plane. The main focus of this section is to extend the usual rectangular coordinate system in the xy-plane into a 3 dimensional space or a 3-space . This is done by introduc- 3-space ing a z-coordinate along a z-axis that is perpendicular to the xy-plane. Then this new xyz-coordinate system is separated into 8 regions called octants where the xyz-coordinate system regions are separated by the xy-, xz- and yz-planes. If x > 0, y > 0 and z > 0 then this is called the first octant . octant The xyz-coordinate system is typically considered to be a right-handed system...
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- Spring '10