- Don't forget to do your on-line Course Evaluations: you should have received an email that looks something like this:
Please read this message and then go to the On-line Student Evaluation Project web site eval.nku.edu. On this site you can log-in to complete evaluations.
At the same web site, you may view a video of University Provost Gail Wells discussing the importance of evaluations and the Online Student Evaluation Project.
For full-term classes, students will be able to answer the evaluation questions (24 hours a day) from Monday, December 1st through Friday, December 12th.
- According to the exam
schedule, you all should be free from 1:00-3:00 on Tuesday,
12/16. I'll plan to hold a review session then (which will include a
chance for you to ask problems about any of the topics we've covered.
- I also plan to hold a review session Thursday, 12/18, in the
evening. Let's say 7:00-9:00 (or until everyone is gone!;)
- HW13.1 Returned:
- #30
- #59: forces!
- Homework pp. 673-674, #13, 18, 22, 26, 45, 50, 51 is due today
- Homework problems pp. 680-683, #2, 7, 8, 14, 16, 41, 42, 60, 70,
71 (due tomorrow).
- A problem or two from 13.4/13.5 will appear on the final. I'll give you a list of possibilities tomorrow.
- 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 useful in three-space, which makes it somewhat unusual (the dot product
exists and is useful in any-dimensional space).
Because the cross product is linear in the components, we can define it on the unit vectors in three space, and then deduce it using the component-wise definition of a three-vector.
Don't worry about the determinant stuff in these sections: we'll leave that to linear algebra! It's an interesting little side-trip at this point, but becomes fundamental in LA.
Examples:
- #20, p. 691
- #28, p. 691
- The force
on a proton moving at velocity
(in m/s) in a uniform magnetic field
(in teslas) is
(in Newtons), where
Coulombs:
- Time for a little calculus, differential equations, and linear algebra:
- Let's see if we can deduce the trajectory of the proton in this illustration.
- #36, p. 692
- Here is a great explanation
of torque:
- Torque is work to create rotation (work in the technical sense: force times the distance through which the force is applied).
- Think about a wrench: if you want to create rotation (twist off a nut),
how should you apply your force?
The nut is at the origin of our system, and we want to create rotation there. So
is the vector from the nut to the position at which we apply the force (where our hand contacts the wrench).
How can you increase your torque?
How is it affected by the angle, vector
- #64, p. 693
- Equations of Planes
- We'll start by thinking about planes through the origin. Thus, the
vectors I discuss in the next few lines are position vectors,
originating from the origin.
- The parallelpiped equation in Theorem 3 above gives us a hint as
to how to write the equation of a plane containing ("spanned by") two
vectors
: We want to find all vectors
living in the space. Those will be the ones such that
Notice that
satisfies the equation of the plane.
Links:
- We'll start by thinking about planes through the origin. Thus, the
vectors I discuss in the next few lines are position vectors,
originating from the origin.
- 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 useful in three-space, which makes it somewhat unusual (the dot product
exists and is useful in any-dimensional space).