Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

THE STRAIGHT LINE 479
are direction components of L. They are, then, proportional
to 1, m, n:
(2) x -Xo =pl, y- Y = pm, z - o =pn,
where the factor of proportionality, p, varies in value as P
changes position.e
Conversely, if P: (x, y, z) is any point other than P0, for
which equations (2) hold for some value of p, i.e. for which
the quantities (1) are proportional to 1, m, n, it follows that
the direction of PoP is that of L and hence that P lies on L.
The coordinates (x0, y0, z0) of the excepted point PO obviously
satisfy (2), when p = 0. Consequently, equations (2) are satisfied by the coordinates of those points and only those points
which lie on L and so they represent L.
Instead of equations (2), we can write
(3) X - - Y - Yo0 - Z
I m n
We call this continued equality the equations of the line,
remembering from ~ 1 that, of the three equations which in
general result from it, one is superfluous.
If one of the denominators in (3) is zero, so is the corresponding numerator; thus, if n = 0, then z - z = 0. For,
equations (3) are but an abbreviated form of the equations of
proportionality t (2), and if n = 0 the last equation in (2)
reduces immediately to z - zo = 0.
Suppose, for example, that the line is to go through the
point (3, 4, - 6) and have 0, 5, 3 as its direction components.
Here 1 =0, and hence x-x=O0, i.e. x —3 =0. Thus,
X From (2), poP2 = p2 (12 + m2 + n2),
or p =- PoP
/12 + m2 + n2
That is, the numerical value of p is proportional always to the distance
of P from Po. The sign of p depends on the side of Po on which the
point P is situated.
t Cf. Ch. XVI, ~ 9, eq. (5).