Solution to Exercise 3.7.
Anyone
who cannot cope with mathematics is not fully human. At best he is a tolerable
subhuman who has learned to wear shoes, bathe and not make messes in the house.
—
Robert A. Heinlein Time Enough for Love (1973)
Exercise 3.7. Given the line [4, –2, 3].
(a) Write two other sets of homogeneous
coordinates for the line.
Since a line is an equivalence
class, the general form of the line is [4k,
–2k, 3k].
Several possible coordinates for the line are [2, –1, 3/2], [8, –4, 6], [12, –6,
9], etc.
(b) Write an equation of the line.
The equation of the line is or 4x1 – 2x2
+ 3 = 0.
(c) Find two distinct points (x1, x2,
1) on the line.
Solving the equation of part (b), we obtain the following possible general forms:
or
Several examples of points on the line are
etc.
