How to scale arrows to approximately fill a plot region?

In a variety of graphic applications, I plot some data, together with
arrows representing variables
or linear transformations of variables as vectors in the same space, as
in a biplot.

In my applications, the scale of the arrows is arbitrary -- all that
matters is relative length.
I'd like to blow them up or shrink them to fit the available space in
the plot.
The origin is typically at some mean for (x,y), but that is not
necessarily so.
There must be some general, perhaps approximate solution for this problem,
but I can't see it.

Below is a simple test case. I found the approximate solution, scale <-
14 by trial and error.
[I'm ignoring aspect ratio, because that determines the bbox I calculate
from the plot.]


set.seed(123135251)
x <- 2 + 5* rnorm(50)
y <- 5 + 2* rnorm(50)
plot(x,y)

# get bounding box of plot region, in data coordinates
bbox <- matrix(par("usr"), 2, 2, dimnames=list(c("min", "max"),c("x",
"y")))
# center vectors here
origin <- colMeans(bbox)
points(origin[1], origin[2], pch=16, cex=2)

# vectors to be displayed in this space
vectors <- cbind( runif(5), (runif(5)-.5))

# draw arrows, given origin and length in xy
Arrows <- function(xy, lenxy, length=.1, angle=10, ...) {
arrows(xy[1], xy[2], xy[1]+lenxy[,1], xy[2]+lenxy[,2], length=length,
angle=angle, ...)
}

## How to determine scale so that vectors ~ fill the bounding box???
scale <- 14
Arrows(origin, scale*vectors)