Day 29 in Mat228

Today:

• Announcements
  • Homework 13.2 returned
    • Grader: "Some people think that two vectors can point in the same direction een if they are not parallel!"
    • Appeared to be a struggle on #45, p. 674.
  • Sorry: your quizzes ended up at home; I'll have them on Thursday for you.
  • Remember to look over the exam again, correcting your errors, and see me for 40% of your missed points back.
  • Reminder: we'll have a review session Monday evening, prior to the final. This will be question and answer, 7:00-9:00, in this room (if we need to move, I'll post a sign). Let me know if you'll be coming late....
  • There will be a quiz Thursday over the reading.
  • Homework for section 13.3 is due Thursday (I'll post it on my door Friday).

  • It's time for course evaluations. Please take the time to visit eval.nku.edu, and give me your feedback. Thanks!

• Section 13.3: The Dot Product
  • We have looked at examples like #58, p. 665:
    • First and foremost: Newton's laws of motion (in particular ), and statics.
    • The parallelogram rule
    • We can think of the force vectors on the cables as being composed of vector components (as well as coordinate components).

    It turns out that we could see the solution very easily; calculating the force vectors was a little more complicated.

    Now let's think a little more about how we resolve the vector forces into components: we'll find it convenient to define a product of two vectors as follows:

    

    (how would you define this for vectors in two-space?). It's just component-wise multiplication, and we add up the results.

    examples:

    • #1, p. 680 (and let's practice drawing the line connecting the tips of these two position vectors)
    • #6, p. 680
    • #11, p. 681

    It turns out that

    

    where

    

    Otherwise, the angle is acute.

    examples:

    • #23, p. 681
    • Now: here's problem that really shows the power of this concept: #60, p. 681

    Finally then, we can put it all together to get this:

  • Additional examples:
    • #53, 54, p. 681
    • #70, p. 682 (bad picture -- looks like vector F can't make up its mind!)

• Section 13.4: The Cross Product
  • The next product we'll find convenient to define is the cross product, which is a product of two vectors resulting in a third vector, which juts into the space perpendicular to which the two vectors live.

    It is only defined (and useful) in three-space, which makes it somewhat unusual (the dot product exists and is useful in any dimensional space).

    The cross product is linear in its components: that is,

    

    This means that we can define it on the unit vectors in three space, and then deduce it using the component-wise definition of a three-vector.

    

    

    

    

    When you reverse direction, however, you get the negative of the clockwise results:
    

    

    

    Furthermore, if you cross a vector with itself you get the zero vector:
    Example:
    • #20, p. 691

    Now: here are the important geometric (rather than simply algebraic) properties of this product:

    

    Notice the difference in length between the dot- and cross-products:
    for the dot-product, it's the cosine; for the cross-product, it's the sine.

    

    There's another way of calculating the cross-product, which helps organize the calculation, and you may find it useful:

    

    So let me explain what's going on....

    Example:

    • #20, p. 691

    

    

    Examples:

    • #28, p. 691
    • #46, p. 692
    • The force on a proton moving at velocity (in m/s) in a uniform magnetic field (in teslas) is (in Newtons), where Coulombs:

      

      #36, p. 692

    • Torque: #64, p. 693