Detailed objectives are given in each of the sections listed below. 1. Cartesian Space Coordinates. 2. Displacements, Forces, Velocities and Vectors

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1 Unit 1 Vectors In this unit, we introduce vectors, vector operations, and equations of lines and planes. Note: Unit 1 is based on Chapter 12 of the textbook, Salas and Hille s Calculus: Several Variables, 7th ed., revised by Garret J. Etgen (New York: Wiley, 1995). All assigned readings and exercises are from that textbook, unless otherwise indicated. Objectives Detailed objectives are given in each of the sections listed below. 1. Cartesian Space Coordinates 2. Displacements, Forces, Velocities and Vectors 3. Dot Products 4. Cross Products 5. Lines 6. Planes 7. Geometry by Vector Methods Objective 1 a. plot points in a Cartesian coordinate system, and find the distance between two points and the midpoint of the line segment connecting two points. b. sketch a sphere using the Cartesian space coordinates, given its equation. c. sketch planes parallel to the xy-plane, the yz-plane and the xz-plane in a Cartesian coordinate system. d. demonstrate various conclusions relating to a Cartesian coordinate system in three-dimensional space. Mathematics 365 / Study Guide 1

2 Read Section 12.1, pages Complete problems 3, 5, 9, 11, 15, 19, 23, 29, 37, 39 and 43 on pages Cartesian space coordinates rectangular space coordinates right-handed system distance formula equation of a sphere plane symmetry midpoint formula Before you proceed to Objective 2, make certain that you can meet each of the sub-objectives listed under Objective 1. Objective 2 a. find the vector defined by two points and determine the norm of the vector. b. add two vectors using the parallelogram law and using tail-to-head addition. c. multiply a non-zero vector by a non-zero scalar. d. represent a non-zero vector in the xy-plane in terms of its magnitude and the angle it makes with the positive x-axis. e. demonstrate various conclusions relating to vectors defined in a Cartesian coordinate system. 2 Calculus Several Variables

3 Read Section 12.2, pages Complete problems 1, 3, 9, 13, 15, 20, 25, 27, 29, 33, 36, 39, 47 and 51 on pages displacement total displacement Newton s second law of motion force resultant or total force vector scalar zero vector parallel vectors norm of a vector norm properties triangle inequality unit vector Before you proceed to Objective 3, make certain that you can meet each of the sub-objectives listed under Objective 2. Mathematics 365 / Study Guide 3

4 Objective 3 a. define the dot product and interpret it geometrically. b. use the dot product to determine the angle between two vectors, to determine whether two vectors are perpendicular to one another, to determine projections and components of vectors, to find direction angles and direction cosines, and to prove the triangle inequality. c. demonstrate various conclusions relating to the dot product and the triangle inequality. Read Section 12.3, pages Complete problems 3, 5, 7, 11, 13-17, 19, 29-33, 35, 37, 39, 41, 45, 47 and 49 on pages dot product scalar product of a and b properties of the dot product perpendicular vectors law of cosines projection of a vector (e.g., proj b a ) component of a vector (e.g., comp b a ) direction angle direction cosine Schwarz s inequality 4 Calculus Several Variables

5 Before you proceed to Objective 4, make certain that you can meet each of the sub-objectives listed under Objective 3. Objective 4 a. define the cross product and interpret it geometrically. b. determine the cross product of vectors and combinations of vectors. c. determine the area of a triangle and the volume of a parallelepiped. d. find unit vectors that are perpendicular to two given vectors. e. prove various identities involving dot and cross products. f. demonstrate various conclusions related to the cross product. Read Section 12.4, pages Complete all of the odd-numbered problems 1-45 on pages cross product vector product of a and b direction of a b magnitude of a b properties of the cross product anticommutative scalar triple product components of a b Mathematics 365 / Study Guide 5

6 determinant [Note: The textbook directs you to page A-2; however, the determinant is discussed in Section A.2, which begins on page A-6 of the Appendices.] cross product identities Before you proceed to Objective 5, make certain that you can meet each of the sub-objectives listed under Objective 4. Objective 5 a. express a line as a vector parametrization, as a scalar parametrization, and in symmetric form. b. use vectors to determine whether two lines intersect, and if so, to identify the point at which they do so. c. use vectors to find the distance from a point to a line. d. demonstrate various conclusions relating to vector parametrizations of lines. Read Section 12.5, pages Complete problems 1, 3, 5, 7, 11, 13-17, 19, 29-33, 35 and 37 on pages position vector radius vector scalar parametric equations symmetric form 6 Calculus Several Variables

7 intersecting lines parallel lines skew lines perpendicular lines Before you proceed to Objective 6, make certain that you can meet each of the sub-objectives listed under Objective 5. Objective 6 a. express a plane as a scalar equation and as a vector equation. b. find unit normals for a plane. c. determine whether two planes intersect, and if so, find both the angle of intersection and a vector parametrization of the line formed by the intersection. d. find an equation for a plane defined by three noncollinear points. e. find the distance from a point to a plane. f. demonstrate various conclusions relating to planes. Read Section 12.6, pages Complete all of the odd-numbered problems 1-51 on pages scalar equation of a plane normal vector Mathematics 365 / Study Guide 7

8 vector equation of a plane collinear points collinear vectors coplanar points coplanar vectors unit normals parallel planes intersecting planes noncollinear points Before you proceed to Objective 7, make certain that you can meet each of the sub-objectives listed under Objective 6. Objective 7 use vectors to prove various geometric statements. Read Section 12.7, pages Prove each of the propositions presented in Section 12.7, using vector methods. Before you proceed to Unit 2, make certain that you can meet Objective 7. 8 Calculus Several Variables