Homework for Symmetries

  1. (CircleVector)

    Consider the geometry of $\vert \rr-\rrp\vert$.

    1. Make a sketch of the graph $$ \vert \Vec r - \Vec a \vert = 2 $$ for each of the following values of $\Vec a$: $$ \begin{eqnarray} \Vec a &=& \Vec 0\\ \Vec a &=& 2 \hat \imath- 3 \hat \jmath\\ \Vec a &=& \hbox{points due east and is 2 units long} \end{eqnarray} $$

    2. Derive a more familiar equation equivalent to $$ \vert \Vec r - \Vec a \vert = 2 $$ for arbitrary $\Vec a$, by expanding $\Vec r$ and $\Vec a$ in rectangular coordinates. Simplify as much as possible. (Ok, ok, I know this is a terribly worded question. What do I mean by “more familiar”? What do I mean by “simplify as much as possible”? Why am I making you read my mind? Try it anyway. Real life is not full of carefully worded problems. Bonus points to anyone who can figure out a better way of wording the question that doesn't give the point away.)

    3. Write a brief description of the geometric meaning of the equation $$ \vert \Vec r - \Vec a \vert = 2 $$

  2. (Tetrahedron)

    Using a dot product, find the angle between any two line segments that join the center of a regular tetrahedron to its vertices. Hint: Think of the vertices of the tetrahedron as sitting at the vertices of a cube (at coordinates (0,0,0), (1,1,0), (1,0,1) and (0,1,1)—you may need to build a model and play with it to see how this works!)

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