Day 30 in Mat228

Last time: 13.4

Next time: Final!

Today:

Announcements
- Return of quiz over 13.3
  - The dot product detects perpendicular vectors: the dot product is zero.
  - The dot product detects parallel vectors: the dot product is the product of their lengths.
- There will be a quiz today over the reading for 13.4.
- Remember to look over the exam again, correcting your errors, and see me for 40% of your missed points back.
- Homework for section 13.3 is due today (I'll post it, and your quizzes, on my door tomorrow morning).
- Final exam:
  - Tuesday, 1:00 p.m.
  - Your final is comprehensive.
  - It will be 40% recent material (chapter 13) and 60% will be material covered in our first two exams.
  - A good strategy is to review your exams (for the older material), and the problems I assigned for homework.
  - 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....
- It's time for course evaluations. Please take the time to visit eval.nku.edu, and give me your feedback. Thanks!
Section 13.4: The Cross Product
- 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,
  
  ${(\bf{v}_1+\bf{v}_2)\times(\bf{w}_1+\bf{w}_2)}=\bf{v}_1\times\bf{w}_1+\bf{v}_1\times\bf{w}_2+\bf{v}_2\times\bf{w}_1+\bf{v}_2\times\bf{w}_2$
  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.
  
  ${\hat{i}\times\hat{j}=\hat{k}}$
  ${\hat{j}\times\hat{k}=\hat{i}}$
  ${\hat{k}\times\hat{i}=\hat{j}}$
  
  When you reverse direction, however, you get the negative of the clockwise results:
  
  ${\hat{i}\times\hat{k}=-\hat{j}}$
  ${\hat{j}\times\hat{i}=-\hat{k}}$
  ${\hat{k}\times\hat{j}=-\hat{i}}$
  
  Furthermore, if you cross a vector with itself you get the zero vector:
  
  ${\hat{i}\times\hat{i}=\hat{j}\times\hat{j}=\hat{k}\times\hat{k}=\bf{0}}$ 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 with these so-called "determinants"
  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
    - Uniform magnetic field means that a particle experiences the same B-vector at every location.
    - What can we say about the acceleration in the direction of B? Does the speed in the B-direction change?
- Application of the Cross-Product (see section 13.5 for more details): Equations of Planes
  - A few assignment problems:
    - pp. 699--, #11, 13, 21, 31
    To motivate you to look at these, a problem like one of these will appear on the final!
  - We'll start with planes through the origin: general planes are just displaced versions of these.
  - Thus, the vectors I discuss in the next few lines can be considered position vectors, originating from the origin.
  - The parallelpiped equation in Theorem 3
    
    gives us a hint as to how to write the equation of a plane containing ("spanned by") two vectors ${\underline{\bf{v}}}$ and ${\underline{\bf{w}}}$ : We want to find all vectors ${\underline{\bf{u}}}$ living in the space. Those will be the ones such that
    
    ${\underline{\bf{u}}\cdot(\underline{\bf{v}}\times\underline{\bf{w}})=0}$
    I.e., ${\underline{\bf{u}}}$ creates a zero volume parallelpiped, because it's living in the plane of ${\underline{\bf{v}}}$ and ${\underline{\bf{w}}}$ .
    This gives us our first equation of a plane: if the coordinates of ${\underline{\bf{u}}}$ are given by ${\langle{x,y,z}\rangle}$ and the coordinates of ${\underline{\bf{v}}\times\underline{\bf{w}}}$ are given by ${\langle{a,b,c}\rangle}$ (that is ${\langle{a,b,c}\rangle}$ is a normal vector) then the equation becomes
    
    ${ax+by+cz=0}$
    This is one form of the equation of a plane (through the origin). We say that the cross product ${\langle{a,b,c}\rangle}$ is normal to the vectors that live in the plane (or normal to the plane): that is, that it is perpendicular to the plane.
    Notice that we can instantly see a normal vector to the vectors in the plane in this form: just pick off the coefficients of the variables (a, b, and c) and turn those into a normal vector.
    
    Now (for the second equation!): notice that ${\underline{\bf{u}}=s\underline{\bf{v}}+t\underline{\bf{w}}}$ satisfies the equation of the plane ${\underline{\bf{u}}\cdot(\underline{\bf{v}}\times\underline{\bf{w}})=0}$ , since ${\underline{\bf{v}}}$ and ${\underline{\bf{w}}}$ are both perpendicular to their cross product. Hence we have another (parametric) equation of a plane (through the origin):
    ${\underline{\bf{u}}(s,t)=s\underline{\bf{v}}+t\underline{\bf{w}}}$
  - To offset these planes from the origin, one needs only reinterpret in the two forms of the equation:
    1. ${\underline{\bf{u}}=\underline{\bf{r}}-\underline{\bf{r}}_0=\langle{x,y,z}\rangle-\underline{\bf{r}}_0}$
      which leads to
      
      ${(\langle{x,y,z}\rangle-\underline{\bf{r}}_0)\cdot\langle{a,b,c}\rangle=0}$ or ${ax+by+cz=d}$
      where ${d=\underline{\bf{r}}_0\cdot\langle{a,b,c}\rangle}$
    2. ${\underline{\bf{u}}=s\underline{\bf{v}}+t\underline{\bf{w}}+\underline{\bf{r}}_0}$
      which leads to
      
      ${\underline{\bf{u}}(s,t)=s\underline{\bf{v}}+t\underline{\bf{w}}+\underline{\bf{r}}_0}$
  - Examples:
    - #12, p. 699
    - #29, p. 700
    - #22, p. 700
    - #30, p. 700

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