<<echo=FALSE, results=hide>>=
p <- c(sample(1:3, 1), sample(1:5, 1))
q <- c(sample((p[1] + 1):5, 1), sample(1:5, 1))
d <- abs(p - q)
sol <- round(c(sum(d), sqrt(sum(d^2)), max(d)), digits = 3)
@

\begin{question}
Given two points $p = (\Sexpr{p[1]}, \Sexpr{p[2]})$ and
$q = (\Sexpr{q[1]}, \Sexpr{q[2]})$ in a Cartesian coordinate system:
\begin{answerlist}
  \item What is the Manhattan distance $d_1(p, q)$?
  \item What is the Euclidean distance $d_2(p, q)$?
  \item What is the maximum distance $d_\infty(p, q)$?
\end{answerlist}
\end{question}

\begin{solution}
The distances are visualized below in green ($d_1$), red ($d_2$),
and blue ($d_\infty$).

<<echo=FALSE, results=hide, fig=TRUE>>=
par(mar = c(4, 4, 1, 1))
plot(0, type = "n", xlim = c(0, 6), ylim = c(0, 6), xlab = "x", ylab = "y")
grid(col = "slategray")
if(d[1] >= d[2]) {
  lines(c(p[1], q[1]), c(q[2], q[2]) - 0.05, lwd = 2, col = "darkblue")
} else {
  lines(c(p[1], p[1]) - 0.05, c(p[2], q[2]), lwd = 2, col = "darkblue")
}
lines(rbind(p, q), lwd = 2, col = "darkred")
lines(c(p[1], p[1], q[1]), c(p[2], q[2], q[2]), lwd = 2, col = "darkgreen")
points(rbind(p, q), pch = 19)
text(rbind(p, q), c("p", "q"), pos = c(2, 4))
@

\begin{answerlist}
  \item $d_1(p, q) = \sum_i |p_i - q_i| = |\Sexpr{p[1]} - \Sexpr{q[1]}| +
    |\Sexpr{p[2]} - \Sexpr{q[2]}| = \Sexpr{sol[1]}$.
  \item $d_2(p, q) = \sqrt{\sum_i (p_i - q_i)^2} = \sqrt{(\Sexpr{p[1]} -
    \Sexpr{q[1]})^2 + (\Sexpr{p[2]} - \Sexpr{q[2]})^2} = \Sexpr{sol[2]}$.
  \item $d_\infty(p, q) = \max_i |p_i - q_i| = \max(|\Sexpr{p[1]} -
    \Sexpr{q[1]}|, |\Sexpr{p[2]} - \Sexpr{q[2]}|) = \Sexpr{sol[3]}$.
\end{answerlist}
\end{solution}

%% META INFORMATION
%% \extype{cloze}
%% \exsolution{\Sexpr{sol[1]}|\Sexpr{sol[2]}|\Sexpr{sol[3]}}
%% \exclozetype{num|num|num}
%% \exname{Distances}
%% \extol{0.01}
