Day 24 in Mat228

Last time: More 2-d vectors

Next time: The dot (inner) product

Today:

Announcements
- Homework from 11.5 (randomly graded 2, 4, 33, 43, 45), and 11.6 returned.
  RE 11.6: the grader noted that you're not checking endpoints for convergence. The Radius of convergence is one thing, but then you need to check the endpoints to complete the problem.
  The grader noted that most of you had no idea on #42, so it was good that Darrel asked about that last time!
- Homework from 13.1 is due today.
- Homework from 13.2 is technically due next time, but we'll make it due next Tuesday.
- We'll have our second exam Thursday. It will cover the rest of chapter 11 (11.2-11.7). You may have a 3x5 card with whatever you like on it during the exam.
- Read Section 13.3 for next Tuesday responding to the worksheet for section 13.3. There will be a quiz next Tuesday over the reading.
Any questions on old stuff, as we prepare for the 2nd exam?
We were finishing some examples from section 13.1:
- More physics:
  - #58, p. 665
    - First and foremost: Newton's laws of motion (in particular ${\underline{\bf{F}}=m\underline{\bf{a}}}$ ), 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).
  - #60, p. 665
  - Let's use a little calculus and the parameterization of the position vector in uniform circular motion given in one of our videos
    
    ${x(t)=R\cos(\frac{2\pi}{T}t)}$ ${y(t)=R\sin(\frac{2\pi}{T}t)}$
    
    to derive the velocity and acceleration vectors, and check the video's claims.
Additional material from section 13.2: three-dimensional vectors
- The right-hand rule
- Vocabulary:
  - coordinate planes (p. 666)
  - octants
  - translates/translation (p. 669)
  - parallel vectors
  - equivalent vectors
  - standard basis vectors
- Distance formula in R³:
  
  examples:
  - #1, p. 673
  - #2, p. 673
  - #14, p. 673
- Equations of Spheres and Cylinders
  What are a sphere and a right cylinder?
  - A sphere: all points equidistant from a given center.
  - A right cylinder: all points equidistant from a given line.
  examples: Draw the following:
  - The set of points satisfying
    ${(x-1)^2+(y+1)^2=4,\qquad{-1\le{z}\le{1}}}$
  - The set of points satisfying
    ${(x-1)^2+(y+1)^2+(z-1)^2=4,\qquad{0\le{y}}$
- Parametric Equations of Lines:
- Additional examples:
  - #27, p. 674
  - #39, p. 674
  - #51, p. 674

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