Class Notes, Fall 2001

Week of	Aug 22	Sep 5	Oct 1	Nov 5	Phy 205 page
	Aug 27	Sep 10	Oct 8	Nov 12
		Sep 17	Oct 15	Nov 19
		Sep 24	Oct 22	Nov 26
			Oct 29		Notes for Fall, 2000

Phy 205, Week 2

27 Aug How to multiply vectors by each other. There are two ways that will show up in this course, the first one is called the "scalar product" or "dot product" and the second one is the "vector product" or "cross product". The names come from the way that they're written, as

-> A   · -> B          and        -> A   x -> B

The first of these is a scalar so that the output is just a number (with units). The second will require a direction to specify it too.

-> A   · -> B   = A B cos θ

where θ is the angle between the two vectors. This odd-looking definition certainly isn't obvious, and it probably took a while to work out when it was first invented sometime in the nineteenth century. If you look at the simplest possible case, that of one dimension, you hardly need vectors, but you can see what this is in that special case. It looks just like the multiplication of ordinary numbers in which a positive times a positive is positive and a positive times a negative is negative.

This definition has the property that if two vectors are perpendicular then their dot product is zero. cos ( 90^o ) = 0. Other properties of numbers work for this product however, the commutative law and the distributive law apply.

I did an example showing how to use the scalar product to compute angles between vectors. Also how to derive the law of cosines for a triangle.

A demo using an electron gun and a magnet: why is there such a thing as a vector product? Finally state a brief but not yet complete definition of the vector product.

29 Aug Relative velocity. How do you perceive the wind velocity if you are moving? Do a few simple cases to get to the final result.

Demo: the bicycle wheel gyroscope to see that it moves in a direction perpendicular to what you would expect.

Do the vector product in detail, magnitude and direction. Explain the right-hand rule a couple of different ways. Compute such a product using components. Show how this cross product can be interpreted as the area of the parallelogram between the two vectors. Show how to derive the law of sines from the cross product.

The cosine of the difference of two angles is an application of a dot product, and I did it completely. The analogous one for the sine uses the cross product, but I didn't do that one.

31 Aug Do a couple of the homework problems and show some more about the geometric interpretation of the vector and scalar products.

The idea of velocity. Just what is it? All of this discussion will be for the one-dimensional case for now, though the three-dimensional case will follow soon enough.