If it is not already on your hard drive, you will need to download the free DPGraph Viewer to view some of the pictures linked to on this page.

QuickTime free download.

Create Your Formula Sheet In Advance

	Seventh Edition Notes Page
	Allegany College of Maryland has a Calculus III course online.  Unit 1 includes vectors, lines, and planes.  Click on Unit 1 at the top of the page when you get there.
	Online Calculus III  from the Springfield Technical Community College distance learning project is another place you might find help.
	Here are some terrific class notes for Calculus III from Paul Dawkins at Lamar University.
	Here is the HMC introductory tutorial on elementary vector analysis.
	Hotmath  You can look at solutions to problems in exercise sets from a wide variety of mathematics textbooks including Calculus by Larson, Hostetler, Edwards, sixth , seventh, and eighth editions.  They have chapters 1 - 12 available.  Only a few solutions are still free (solutions to problems 15, 25, 35 in each section are free).
	Examples
	Lake Tahoe CC Examples
	Interactive Math Programs  Here are some terrific tools for multivariable calculus and differential equations from Dartmouth.
	DPGraph Pictures:  Two Quadric Surfaces, Quadric Surfaces 2, Little Bang, Little Bang2
	Maple will be used extensively in this course and will be available in class and during exams.  The student version is available for purchase at a reduced rate.  Click here for purchase information.  It will not be required that you purchase your own copy of Maple but highly recommended since it is a wonderful tool and very helpful in this course and Differential Equations.
	Here is a Maple Worksheet with differentiation and integration examples.
	Here is a Maple Worksheet with dot product and cross product examples.

Section 11.1:  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 p771: 63-68.  Note also p771: 69-74, 82.

A vector in the plane in standard position (initial point at the origin) is shown on the right.  This vector (call it u) can be represented in terms of its magnitude and the angle A that it makes with the positive x-axis as shown below.  Click the picture to enlarge.

In general

Section 11.2:  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, p774 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 11.3:  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 784 (and also direction angles), its vector component along another vector (projection), and its vector component orthogonal to another vector as shown on page 785.  You should be able to apply the definition of work given on p787.  Some good exercises to look at would be pp788-9: 47-50, 71.  Dot Product Applet (geometric interpretation)

Direction Cosines

Section 11.4:  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 11.5.  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 pp796-7: 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.

In the animation on the right,  v1 = <1,3,-5>, v2 = <-3,4,6>,  and v3 = v1 x v2 = <-7,4,-6>.

Here is a Maple Worksheet with dot product and cross product examples.

Section 11.5:  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 pp805-7: 3-30, 35-62, 75, 76, 81-100 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).

x = -2 + 3t,    y = 2 + t,     z = -4 - t

2x - 3y - z + 4 = 0

Sections 11.6 and 11.7 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 11.6.  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.  In this Maple Worksheet you can see the constructions of a variety of surfaces in space.   Here is a link to some surfaces of revolution on my Calculus II site along with volume and surface area computations.  Here is another Maple Worksheet relating to a surface of revolution and two potential generating curves.

 

x2 + s*y2 + (1-s)z2 = 1   with s varying between 0 and 1

2x + 3y + 4z = 24

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 a picture of the vase and the data for the volume and surface area extra credit problem.

Here is an introduction to 11.7, spherical and cylindrical coordinates.  This applet might help to introduce you to spherical coordinates.

In chapter 12 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 12.

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        Lane Vosbury, Mathematics, Seminole State College   email:  vosburyl@seminolestate.edu

        This page was last updated on 08/21/14          Copyright 2002          webstats