Lesson plan: Vector Reflections and Flips

November 16, 2009 3 Comments

Objectives:

Show that a 180^\circ flip of the vector \mathbf a about the vector \mathbf m is \mathbf a'=\mathbf e_{\mathbf m}\mathbf a\mathbf e_{\mathbf m}, where \mathbf e_{\mathbf m} is the unit vector along \mathbf m.

Methodology

  1. Draw the vectors \mathbf a, \mathbf m, \mathbf m, and \mathbf a in the same plane.
  2. Show that the magnitude of the dot and wedge products of \mathbf a and \mathbf m are the same as those of \mathbf m and \mathbf a'.
  3. Show that \mathbf a'\mathbf m=\mathbf m\mathbf a.
  4. Show that the multiplicative inverse of a vector \mathbf m is \mathbf m^{-1}=\mathbf m/|\mathbf m|^2.
  5. Show that \mathbf a'=\mathbf e_{\mathbf m}\mathbf a\mathbf e_{\mathbf m}, where \mathbf e_{\mathbf m}=\mathbf m/|\mathbf m
  6. Give an example of \mathbf a and \mathbf m in the \mathbf e_1\mathbf e_2 plane.

Evaluation

  1. The students have difficulty understanding the multiplicative inverse of a vector, even if they know that \mathbf a\mathbf b=\mathbf a\cdot\mathbf b+\mathbf a\wedge\mathbf b.  To help them understand, I showed that \mathbf a^2=\mathbf a\cdot\mathbf a=|\mathbf a|^2, because \mathbf a\wedge\mathbf a=0.  The definition of the multiplicative inverse of \mathbf a immediately follows.
  2. In the evaluation of vector products for reflections, I stumbled upon the identity (\mathbf e_1+\mathbf e_2)(\mathbf e_1-\mathbf e_2)=2\hat i\equiv 2\mathbf e_1\mathbf e_2.
  3. The discussion is still unfinished.

Duration: 50 minutes.

http://www.prolifeworldcongress.org/zaragoza2009/index.php?option=com_content&task=view&id=173&Itemid=104

Filed under Uncategorized Tagged with ,

About Quirino M. Sugon Jr
Theoretical Physicist in Manila Observatory

3 Responses to Lesson plan: Vector Reflections and Flips

  1. hector says:
    December 15, 2009 at 3:41 pm

    Great post! Please keep posting this Lesson Plans.

  2. peeterjoot says:
    March 12, 2010 at 1:48 pm

    If the students have matrix formalism under their belts, you could also point out that the moore-penrose inverse of a column vector:

    v^{-1} = v / {\lVert v \rVert}^2,

    takes the same form as the clifford vector inverse.

    Now, if the students haven’t seen least squares and this MP non-square matrix inversion method, then this is probably not helpful. It seems to me that there is a deep connection there (but I haven’t finished exploring it).

  3. peeterjoot says:
    March 12, 2010 at 1:50 pm

    oops. meant to have a transpose in there (and should have noted that the MP inverse is only one sided).

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