Homework:  Read Sections 10.1 - 10.3, and work the following problems: (You do not need to hand these problems in, but any of them may appear on the quiz on Tuesday 8/29/00.)

• Section 10.1: 7; 13a,b; 21, 24, 29, 33, 39a,b,c,d,e,f; 41, 44, 49, 52, 57, 63(can just use calculator), 84-89
• Section 10.2: 1, 7, 17, 23, 25, 31, 34, 45, 51, 54, 62, 85
• Section 10.3: 1, 9, 16, 24, 25, 31, 32, 45, 48, 62, 65

Overview:  Vectors will play a major role throughout this course (as well as in Calc IIIb). It is essential that you master them right away. Some key points:

• Vectors are ordered lists of numbers that can be used to represent a variety of things:
  • points in the plane or in space, (perhaps representing the position of an object)
  • directed line segments, (representing for example velocity, acceleration, force, etc).
• NOTE: The text talks about vectors only as directed line segments. This is the way physicists typically use vectors, but I find this to be cumbersome at times-- particularly when we discuss space curves and vector valued functions. So be aware that I will treat vectors slightly differently than the text does.
• We can associate a direction and magnitude to any vector. You should know how to calculate these.
• You can add two vectors together and you can multiply vectors by scalars--this is all very intuitive, and should be easy for you to master.
• Page 703 gives some geometric interpretations of vector addtion and scalar multiplication. These pictures will be very important later on.
• You can also multiply two vectors together, but this is NOT obvious how to do. In fact, there are two different definitions of multiplication (DOT product and CROSS product). We are studying the DOT product this week. You must learn special rules for how to compute the DOT product, and you will need to practice doing it until it is second nature. IT IS ESSENTIAL THAT YOU MASTER DOT PRODUCTS THIS WEEK, OR YOU WILL STRUGGLE LATER ON!!!
• DOT products are useful for determining the angle between two vectors. Memorize the formula in Theorem 10.5, but more importantly remember this: TWO VECTORS ARE ORTHOGONAL IF THEIR DOT PRODUCT IS ZERO.
• The relationship between DOT products and angles plays a key role in deriving the formula for deriving the formula in Theorem 10.6 for the projection of one vector onto another. We'll use this formula a lot this semester, so (this sounds like a broken record), but memorize this formula now and practice using it--you'll be glad you did later on.

• Distance formula.
• Equation of a sphere. (Focus on the intuition behind this formula)

What you should know by the end of this week:

Terminology:  Component form of a vector, directed line segment, norm, vector sum, scalar multiple, unit vector, standard unit vectors (i,j,k), linear combination, right-handed, coordinate system, zero vector, equality of vectors, parallel vectors, orthogonal vectors, dot product, projection, vector components.

1. Given a directed line segment, find the component form of the vector it represents.
2. Find the length and direction (angle with the x-axis) of a given vector in component form.
3. Find the component form of a vector of a given length and direction.
4. Draw a picture illustrating how to find the sum of two vectors, or what effect scalar multiplication has on a vector.
5. Write a vector as a linear combination of unit vectors.
6. Find the equation of a sphere given two points on either end of a diameter.
7. Determine whether two vectors are parallel.
8. Calculate the dot product of two vectors.
9. Find the angle between two vectors.
10. Find the projection of a vector onto another vector.
11. Find the component of a vector orthogonal to another vector.
12. Find a unit vector in the same direction as a given vector.
13. Use vectors to determine whether three points are collinear.
14. Calculate direction angles in 3-dimensional space.
15. Applications
16. Force (see 10.1, example 7)
17. Velocity (10.1, example 8)
18. Work (calculate the work done by a force acting on a moving object).