Homework:  Read Sections 10.4 - 10.6, and work the following problems: (You do not need to hand these problems in, but any of them may appear on the quiz.) (problems with and asterisk (*) are problems that were assigned in the first problem set)

• Section 10.4:   1, 5, 9, 18, 24, 29, 34, 39, 42*
• Section 10.5:   6, 9, 12, 16, 20 (graph only first parametric line, Gyro is not capable of graphing 2 lines), 25, 33, 38, 44*, 55, 60*, 64, 65
• Section 10.6:   1-6, 9, 10, 12*, 17, 31, 34*, 39 (Try to do problems 9, 10, and 12 without Gyro, then check with Gyro).

Overview:  The goals for this week are i) to understand the cross-product and ii) to get practice working in three dimensional space. The material in sections 10.4 and 10.5 is more important than the material in 10.6. Some comments:

• I find it easier to remember the determinant rule for calculating cross-products. Be sure to practice this enough that it is easy for you--especially if you plan to take Calc IIIb, since we'll use it a lot in chapter 14.
• When studying Thm 10.7 (properties of cross products), pay special attention to what is different that what you would expect from multiplication of scalars.
• Of the four properties of cross products listed in Theorem 10.8, the most important is property 1: u x v is orthogonal to both u and v. This fact will be used several times throughout the course--the first will be in section 10.5 when we look at equations of planes in space.
• In studying equations of lines and planes, (and later surfaces) in space, I find it helpful to distinguish between implicit and explicit descriptions.
  • An implicit description is a set of equations that must be satisfied for every point on the line, plane or surface. For example, the standard equation of a plane given in theorem 10.12 is an implicit equation; every point (x,y,z) on the plane must satisfy this relationship. With an implicit description, it is very easy to check whether or not a given point belongs to the line, plane or surface; but it may be harder to find a point belonging to the line, plane, or surface.
  • An explicit description is a formula for generating points on the line, plane or surface. The parametric equation of a line given in theorem 10.11 is an example of an explicit description. (in contrast, the symmetric equations form of the line is an implicit description). With an explicit description, it is easy to find points belonging to the set, but it is harder to verify whether a given point belongs to the set.
• A line in space is characterized by a point on the line, and a vector parallel to the line. In contrast, a plane is characterized by a point on the plane and a vector orthogonal to the plane.
• The most important thing you can learn from section 10.6 is to use the trace of the surface in a plane to facilitate sketching a surface defined by an equation. With this single skill, you should be able to classify any of the quadric surfaces just by sketching.

Things to learn this week: 

Terminology:  cross product, triple scalar product, parametric equations, symmetric equations, direction vector, standard and general forms of an equation for a plane in three dimensions, cylindrical surface, generating curve, quadric surface, surface of revolution.

Essential Skills (The quiz will emphasize these skills)

1. Calculate the cross-product of 2 vectors in 3 dimensions. ( Practice this until it is easy !!).
2. Use the cross-product to find a vector orthogonal to two given vectors.
3. Use the cross-product to calculate the area of a parallelogram having two given vectors as adjacent sides
4. Find a parametric equation for a line in space that passes through a given point and is parallel to a given vector.
5. Find a parametric equation for a line passing through two given points in space.
6. Find the equation (standard or general form) of a plane in space containing three noncolinear points.
7. Given an equation for a plane, sketch the plane.

Lesser priority:

1. Calculate the triple scalar product of three vectors.
2. Use the triple scalar product to calculate the volume of a parallelpiped with three given vectors as adjacent sides.
3. Calculate the angle between two planes given normal vectors to each plane.
4. Find the line of intersection of two planes.
5. Find the distance between a point and a plane.
6. Find the distance between two planes.
7. Find the distance between a point and a line in space.
8. Match an equation with its 3-D graph (see 10.6: exercises 1-6).
9. Given an equation for a surface, sketch the xy-,xz-, or yz-traces of the surface.

Lowest priority:  (Don't worry about this until you have the above material mastered.)

1. Calculate the moment of a given force about a point.
2. Given an equation for a curve, determine the equation of a surface of revolution generated by revolving the curve
3. Given the equation of a cylinder whose rulings are parallel to a coordinate axis, sketch the surface in space.
4. Classify and sketch a quadric surface, surface of revolution, or cylinder, given its equation.
5. Find a generating curve for a surface of revolution.