Chapter 11 Vectors and the Geometry of Space

Chapter 11 Vectors and the Geometry of Space

Figure 11. 1 and Figure 11. 2 Copyright © Houghton Mifflin Company. All rights reserved. 11 -2

Figure 11. 4 and Definition of Component Form of a Vector in the Plane Copyright © Houghton Mifflin Company. All rights reserved. 11 -3

Definitions of Vector Addition and Scalar Multiplication Copyright © Houghton Mifflin Company. All rights reserved. 11 -4

Figure 11. 6 Copyright © Houghton Mifflin Company. All rights reserved. 11 -5

Figure 11. 7 Copyright © Houghton Mifflin Company. All rights reserved. 11 -6

Figure 11. 8 Copyright © Houghton Mifflin Company. All rights reserved. 11 -7

Theorem 11. 1 Properties of Vector Operations Copyright © Houghton Mifflin Company. All rights reserved. 11 -8

Theorem 11. 2 Length of a Scalar Multiple 11 -9

Theorem 11. 3 Unit Vector in the Direction of v Copyright © Houghton Mifflin Company. All rights reserved. 11 -10

Figure 11. 14 Copyright © Houghton Mifflin Company. All rights reserved. 11 -11

Figure 11. 15 Copyright © Houghton Mifflin Company. All rights reserved. 11 -12

Figure 11. 16 Copyright © Houghton Mifflin Company. All rights reserved. 11 -13

Figure 11. 17 Copyright © Houghton Mifflin Company. All rights reserved. 11 -14

Vectors in Space box Copyright © Houghton Mifflin Company. All rights reserved. 11 -15

Figure 11. 19 Copyright © Houghton Mifflin Company. All rights reserved. 11 -16

Figure 11. 20 Copyright © Houghton Mifflin Company. All rights reserved. 11 -17

Definition of Parallel Vectors and Figure 11. 21 Copyright © Houghton Mifflin Company. All rights reserved. 11 -18

Definition of Dot Product Copyright © Houghton Mifflin Company. All rights reserved. 11 -19

Theorem 11. 4 Properties of the Dot Product Copyright © Houghton Mifflin Company. All rights reserved. 11 -20

Theorem 11. 5 Angle Between Two Vectors and Figure 11. 24 Copyright © Houghton Mifflin Company. All rights reserved. 11 -21

Alternative form of dot product and Figure 11. 25 Copyright © Houghton Mifflin Company. All rights reserved. 11 -22

Definition of Orthogonal Vectors Copyright © Houghton Mifflin Company. All rights reserved. 11 -23

Figure 11. 26 Copyright © Houghton Mifflin Company. All rights reserved. 11 -24

Definition of Projection and Vector Components and Figure 11. 29 Copyright © Houghton Mifflin Company. All rights reserved. 11 -25

Theorem 11. 6 Projection Using the Dot Product Copyright © Houghton Mifflin Company. All rights reserved. 11 -26

Definition of Work and Figure 11. 33 Copyright © Houghton Mifflin Company. All rights reserved. 11 -27

Definition of Cross Product of Two Vectors in Space 11 -28

Theorem 11. 7 Algebraic Properties of the Cross Product Copyright © Houghton Mifflin Company. All rights reserved. 11 -29

Theorem 11. 8 Geometric Properties of the Cross Product Copyright © Houghton Mifflin Company. All rights reserved. 11 -30

Figure 11. 36 Copyright © Houghton Mifflin Company. All rights reserved. 11 -31

Theorem 11. 9 The Triple Scalar Product Copyright © Houghton Mifflin Company. All rights reserved. 11 -32

Theorem 11. 10 Geometric Property of Triple Scalar Product and Figure 11. 41 Copyright © Houghton Mifflin Company. All rights reserved. 11 -33

Figure 11. 43 Copyright © Houghton Mifflin Company. All rights reserved. 11 -34

Theorem 11. 11 Parametric Equations of a Line in Space Copyright © Houghton Mifflin Company. All rights reserved. 11 -35

Figure 11. 45 Copyright © Houghton Mifflin Company. All rights reserved. 11 -36

Theorem 11. 12 Standard Equation of a Plane in Space Copyright © Houghton Mifflin Company. All rights reserved. 11 -37

Figure 11. 47 Copyright © Houghton Mifflin Company. All rights reserved. 11 -38

Figure 11. 49 Copyright © Houghton Mifflin Company. All rights reserved. 11 -39

Figure 11. 50 Copyright © Houghton Mifflin Company. All rights reserved. 11 -40

Figure 11. 51 Copyright © Houghton Mifflin Company. All rights reserved. 11 -41

Theorem 11. 13 Distance Between a Point and a Plane and Figure 11. 52 Copyright © Houghton Mifflin Company. All rights reserved. 11 -42

Theorem 11. 14 Distance Between a Point and a Line in Space and Figure 11. 54 Copyright © Houghton Mifflin Company. All rights reserved. 11 -43

Definition of a Cylinder and Figure 11. 56 Copyright © Houghton Mifflin Company. All rights reserved. 11 -44

Figure 11. 57 and Equations of Cylinders Copyright © Houghton Mifflin Company. All rights reserved. 11 -45

Quadric Surface Copyright © Houghton Mifflin Company. All rights reserved. 11 -46

Ellipsoid Copyright © Houghton Mifflin Company. All rights reserved. 11 -47

Hyperboloid of One Sheet Copyright © Houghton Mifflin Company. All rights reserved. 11 -48

Hyperboloid of Two Sheets Copyright © Houghton Mifflin Company. All rights reserved. 11 -49

Elliptic Cone Copyright © Houghton Mifflin Company. All rights reserved. 11 -50

Elliptic Paraboloid Copyright © Houghton Mifflin Company. All rights reserved. 11 -51

Hyperbolic Paraboloid Copyright © Houghton Mifflin Company. All rights reserved. 11 -52

Figure 11. 62 Copyright © Houghton Mifflin Company. All rights reserved. 11 -53

Surface of Revolution Copyright © Houghton Mifflin Company. All rights reserved. 11 -54

The Cylindrical Coordinate System Copyright © Houghton Mifflin Company. All rights reserved. 11 -55

Figure 11. 66 Copyright © Houghton Mifflin Company. All rights reserved. 11 -56

Figure 11. 69 Copyright © Houghton Mifflin Company. All rights reserved. 11 -57

Figure 11. 70 Copyright © Houghton Mifflin Company. All rights reserved. 11 -58

Figure 11. 74 Copyright © Houghton Mifflin Company. All rights reserved. 11 -59

The Spherical Coordinate System Copyright © Houghton Mifflin Company. All rights reserved. 11 -60

Figure 11. 75 Copyright © Houghton Mifflin Company. All rights reserved. 11 -61

Figure 11. 76 Copyright © Houghton Mifflin Company. All rights reserved. 11 -62