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Section 10.1: Some of the skills involved in this section will be included in problems from later in the chapter. You should understand vector operations like vector addition and subtraction. Of particular note would be recognizing what it means for two vectors to be equal and applying this to a problem like those on p725: 63-68. Note also p725: 69-72, 80.
Section 10.2: You should be able to understand what a three dimensional vector in standard position would look like given its component representation. Be able to find the center of a sphere given its equation (you will need to be able to complete the square). Be able to find the equation of a sphere given sufficient information about it (example 2, p728 would be an example of this). Again, some of the skills involved in this section will be included in problems from later in the chapter. Introduction to 3D vectors Here is an applet for looking at points in space and the associated vector in standard position.
Here is a short Powerpoint presentation on operations and definitions for vectors in space. The operations and definitions would be similar for vectors in the plane. Vectors in two and three dimensions from Duke University. Section 10.3: Here is some material on Vectors in the Plane from Springfield Technical Community College. You should be able to compute and use the dot product of two vectors and the angle between two vectors. Given the component representation of a vector, you should be able to find its direction cosines as shown on page 738 (and also direction angles), its vector component along another vector (projection), and its vector component orthogonal to another vector as shown on page 739. You should be able to apply the definition of work given on p741. Some good exercises to look at would be pp742-3: 45-48, 67, 71, 72. Dot Product Applet (geometric interpretation)
Section 10.4: You should be able to compute and use the cross product of two vectors and know what the resulting cross product would look like. Some of the skills involved in this section will be included in problems from 10.5. You should be able to find areas of triangles and parallelograms and volumes of parallelepipeds given the coordinates of their vertices or vectors determining them. Exercises to look at would be pp750-1: 31-36, 41-48. Cross Product Applet (geometric interpretation) Here are some presentations of dot product and cross product from a slightly different initial perspective.
Here is a Maple Worksheet with dot product and cross product examples. Section 10.5: The majority of the test will come from this section on lines and planes in space. You will need to be able to find the equation representations of lines and planes in space given sufficient information. An interesting variation on this involves the vector equation of a line and the vector equation of a plane which is what I used in deriving the equations of lines and planes for this course. You will need to be able to find the distance between points, lines, and planes (any combination) in space. Here is a development of the vector formula I presented in class for the distance between two skew lines (and also the distance between a point and a plane). You will need to be able to find whether or not two lines intersect, and if so, find the point of intersection and the cosine of the angle of intersection. You need to be able to find the acute angle of intersection of two planes. You need to be able to find the parametric representation of the line of intersection of two planes. You need to be able to find the point of intersection of a plane and a line. Here is a vector representation of the intersection of a plane and a line. Good exercises to look at would be pp759-661: 3-22, 27-50, 63, 64, 67-82 plus distance between two skew lines. Here is the HMC tutorial on lines and planes in space. Here is a powerpoint presentation of examples of finding equations of lines and planes in space (incomplete).
Sections 10.6 and 10.7 will not be on Exam I. They will be included on future exams when they are used. Quadric Surfaces has some nice animations (see below) of some of the surfaces described in 10.6. Here is a DPGraph animation of a surface deforming into a variety of quadric surfaces. You will need to download the free DPGraph Viewer to view it. In this DPGraph picture of Quadrics you can use the scrollbar to activate "a" . By using the scrollbar to vary the value of "a" you can see the various quadric surfaces shown in the animation. The equation being graphed is given below.
Here are some DPGraph pictures of a cylinder, a surface of revolution, and Quadric surfaces: Cylinder, Surface of Revolution, Surface of Revolution 2, Ellipsoid, Hyperboloid(both), Elliptic Cone, Paraboloid (both). You can click on the pictures below to see animations.
Click here to see how you could use a surface of revolution to model a goblet (and other things). There is an Extra Credit opportunity at the end of my demonstration. EXTRA CREDIT: Find the volume and surface area of the vase modeled by the data points given.
Here is an introduction to 10.7, spherical and cylindrical coordinates. This applet might help to introduce you to spherical coordinates. In chapter 11 we will be studying vector valued functions. You will need to review the parametric representation of a curve in preparation for this chapter. You have actually already seen in this course the parametric representation of lines in space. This introduction to parametric vector equations might help prepare you for chapter 11.
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This site contains links to other Internet sites. These links are not endorsements of any products or services in such sites, and no information in such site has been endorsed or approved by this site. Lane Vosbury, Mathematics, Seminole State College email: vosburyl@seminolestate.edu This page was last updated on 08/21/14 Copyright 2002 webstats |
