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\begin{center}
  {\bf University of Kentucky, Physics 306 \\
    Homework \#2, Rev. A, due Monday, 2022-01-30 }
\end{center}

\vspace{.3 in}


{\bf 1. Projections}: relational versus parametric linear/planar geometry.

{\bf \quad a)} Show graphically that the following equations define the set of points $\{\bs x\}$ \mbox{on a {\bf line} or {\bf plane},}
\begin{center}
  \begin{tabular}{r|@{\qquad}l@{\qquad}l}
    & relational & parametric \\ \hline
    line & $\{\bs x ~|~ \bs a \times \bs x = \bs d\}$  &  $\{\bs x = \bs x_1 + \bs a \alpha \phantom{+ \bs c \gamma} ~~|~ \quad \alpha \in \mathbb{R}\}$ \\
    plane& $\{\bs x ~|~ \bs A \cdot \bs x = D\}$  &  $\{\bs x = \bs x_2 + \bs b \beta + \bs c \gamma ~|~ \beta,\gamma \in \mathbb{R}\}$ \\
  \end{tabular}
\end{center}
where $\bs a, \bs d, \bs A, D$ are constants which define the geometry, $\bs x_{1,2}$ are fixed points on the line and plane respectively, and $\alpha, \beta, \gamma$ are parameters that vary along the line/plane (they uniquely parametrize points in the line/plane).  [bonus: Show the 5th relation $\{\bs x = \bs x_2 + \bs A \times \bs \delta ~|~ \bs \delta \in \mathbb{R}^{3}\}$.]

{\bf \quad b)} What constraint between $\bs a$ and $\bs d$ is implicit in the formula $\bs a \times \bs x = \bs d$?  What is the relation between $\bs b$, $\bs c$, and $\bs A$?  Substitute $\bs x$ of each parametrization into its relational equation to show the consistency between the two forms and to derive $\bs d$ and $D$ in terms of $\bs a, \bs x_1$ and $\bs A, \bs x_2$.

{\bf \quad c)} Define $\bt a \equiv \bs A /( \bs a{\cdot} \bs A)$, which is parallel to $\bs A$, and {\it normalized} in the sense that $\bs a \cdot \bt a = 1$.  Using the BAC-CAB rule, show that $\bs x = \bs a (\bt a\cdot \bs x) - \bt a \times (\bs a \times \bs x)$ for all $\bs x$.  Illustrate this non-orthogonal projection of $\bs x$ onto vectors $\bs x_{1}$ parallel to the line plus $\bs x_{2}$ parallel to the plane.  [bonus: use this to calculate the point $\bs x_0$ at the intersection of the line and plane in terms of $\bs a, \bs d, \bs A, D$.  Verify this by showing that $\bs x_0$ satisfies the relational equation for both the line and plane.]

{\bf \quad d)} Let ~$\bt a \equiv \frac{\bv b\times \bv c}{\bv a \cdot \bv b \times \bv c}$, ~$\bt b \equiv \frac{\bv c\times \bv a}{\bv a \cdot \bv b \times \bv c}$,~and ~$\bt c \equiv \frac{\bv a\times \bv b}{\bv a \cdot \bv b \times \bv c}$, where the arrows on $\bs a, \bs b, \bs c$ distinguish them from their {\it covectors} $\bt a, \bt b, \bt c$.  The definition of $\bt a$ is the same as in c) with $\bs A = \bv b \times \bv c$.  Calculate the nine combinations of $(\bv a, \bv b, \bv c)^T \cdot (\bt a, \bt b, \bt c)=I$, i.e.  $\bv a \cdot\bt a=1$, $\bv a \cdot\bt b=0$, \ldots to show they are {\it mutually orthonormal}.  In this sense, $(\bt a, \bt b, \bt c)$ is the \href{https://en.wikipedia.org/wiki/Dual_basis}{\it dual} basis of $(\bv a, \bv b, \bv c)$. [bonus: Show using Cramer's rule that $[\bt a|\bt b|\bt c]$ = $[\bv a|\bv b|\bv c]^{-1}$, ie. that it is a {\it reciprocal basis}.]

{\bf \quad e)} The {\it contravariant components} of $\bs x$ are defined as the components $(\alpha,\beta,\gamma)$ that satisfy the equation $\bs x = \bv a \alpha + \bv b \beta + \bv c \gamma$; i.e., $(\bv a, \bv b, \bv c)$ is the contravariant basis.  Using $\bs x \cdot \bt a$, etc., calculate the three contravariant components of $\bs x$ in terms of dot products.  Likewise, find the {\it covariant components} $(\tilde\alpha,\tilde\beta,\tilde\gamma)$ of $\bs x$, defined as the components $(\tilde\alpha,\tilde\beta,\tilde\gamma)$ which satisfy $\bs x = \bt a \tilde\alpha + \bt b \tilde\beta + \bt c \tilde\gamma$; i.e.  $(\bt a, \bt b, \bt c)$ is the covariant basis.


{\bf 2.} The {\bf \href{https://en.wikipedia.org/wiki/pauli_matrices}{Pauli matrices}}, defined as $\sigma_{x}\equiv\pmatrix{0&1\cr1&0}$, $\sigma_{y}\equiv\pmatrix{0&\!\!\!\!-i\cr i&0}$, $\sigma_{z}\equiv\pmatrix{1&0\cr0&\!\!\!-1}$, are used to describe quantum mechanical spin-1/2 particles.  Show that $\sigma_{j}\sigma_{k}=\delta_{jk}I+i\varepsilon_{jk\ell}\sigma_{\ell}$.  What is the {\it triple product} $\sigma_{x}\sigma_{y}\sigma_{z}$?  Thus $\sigma_{j}$ acts like a basis vector, the product of the Pauli matrices has the structure of the dot, cross, and triple products.

[bonus: Show that product of {\it \href{https://en.wikipedia.org/wiki/gamma_matrices}{Dirac matrices}} is $\gamma^{\mu}\gamma^{\nu}=g^{\mu\nu}I+i\sigma^{\mu\nu}$ and calculate the components $g^{\mu\nu}$ and the matrices $\sigma^{\mu\nu}$.]


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