Graphs Involving the Distance Formula

• AIMS Maxwell 2021

  Learn more about the geometry of \(\vert \vec{r}-\vec{r'}\vert\) in two dimensions.

  1. Make sketches of the following functions, by hand, on the same axes: \begin{align*} y &= \sin x\\ y &= \sin(2+x) \end{align*} Briefly describe the role that the number 2 plays in the shape of the second graph

  2. Make a sketch of the graph \begin{equation*} \vert \vec{r} - \vec{a} \vert = 2 \end{equation*}

     for each of the following values of \(\vec a\): \begin{align*} \vec a &= \vec 0\\ \vec a &= 2 \hat x- 3 \hat y\\ \vec a &= \text{points due east and is 2 units long} \end{align*}

  3. Derive a more familiar equation equivalent to \begin{equation*} \vert \vec r - \vec a \vert = 2 \end{equation*} 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.)
  4. Write a brief description of the geometric meaning of the equation \begin{equation*} \vert \vec r - \vec a \vert = 2 \end{equation*}