Theoretical Mechanics: Fall-2026
HW 1: Due W Day 6 MathBits

1. Angles from the Dot Product

   Let \begin{align*} \boldsymbol{\vec a} &= \boldsymbol{\hat x}-3\boldsymbol{\hat y}-\boldsymbol{\hat z}\\ \boldsymbol{\vec b} &= \boldsymbol{\hat x}+\boldsymbol{\hat y}+2\boldsymbol{\hat z}\\ {\boldsymbol{\vec c}} &= -2\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z}\\ \boldsymbol{\vec d} &= -\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z} \end{align*}

   Use the dot product to determine: Which pairs of these vectors (if any)

   1. Are perpendicular?
   2. Are parallel?
   3. Have an angle less than \(\pi/2\) between them?
   4. Have an angle of more than \(\pi/2\) between them?

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!)

3. Graphs Involving the Distance Formula

   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(x+2) \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*}