```{r data generation, 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)
```

Question
========
Given two points $p = (`r p[1]`, `r p[2]`)$ and
$q = (`r q[1]`, `r q[2]`)$ in a Cartesian coordinate system:

Answerlist
----------
* What is the Manhattan distance $d_1(p, q)$?
* What is the Euclidean distance $d_2(p, q)$?
* What is the maximum distance $d_\infty(p, q)$?


Solution
========
The distances are visualized below in green ($d_1$), red ($d_2$),
and blue ($d_\infty$).

```{r dist, echo = FALSE, results = "hide", fig.path = "", fig.cap = ""}
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))
```

Answerlist
----------
* $d_1(p, q) = \sum_i |p_i - q_i| = |`r p[1]` - `r q[1]`| + |`r p[2]` - `r q[2]`| = `r sol[1]`$.
* $d_2(p, q) = \sqrt{\sum_i (p_i - q_i)^2} = \sqrt{(`r p[1]` - `r q[1]`)^2 + (`r p[2]` - `r q[2]`)^2} = `r sol[2]`$.
* $d_\infty(p, q) = \max_i |p_i - q_i| = \max(|`r p[1]` - `r q[1]`|, |`r p[2]` - `r q[2]`|) = `r sol[3]`$.


Meta-information
================
exname: Distances
extype: cloze
exsolution: `r sol[1]`|`r sol[2]`|`r sol[3]`
exclozetype: num|num|num
extol: 0.01
