Formula for Slope of 3-D Line

Date: 03/12/97 at 10:33:13
From: Cory and Greg
Subject: Slope of three-dimensional line

Thank you for answering our question about finding the length of a 
line in three dimensions.  Now we would like to know the formula to 
find the slope of a three-dimensional line.  We searched through 
textbooks and tried to adapt the formula, but with no success.

Date: 03/12/97 at 12:37:40
From: Doctor Rob
Subject: Re: Slope of three-dimensional line

There is no direct analogue of the idea of slope in two dimensions.
The subject you are discussing is analytic geometry of three 
dimensions. The following facts should help a little.

A linear equation in x, y, and z, such as ax + by + cz = d, is the
equation of a plane, not a line.  Such equations can be put into one
standard form by dividing by sqrt(a^2+b^2+c^2).  The resulting
coefficients of x, y, and z have the property that the sum of their
squares is 1.  Another standard form is gotten by dividing by d, and
writing the equation as x/(d/a) + y/(d/b) + z/(d/c) = 1.  From this
form you can read off the intercepts with the x-, y-, and z-axes:
(d/a, 0, 0) is the x-intercept, (0, d/b, 0) is the y-intercept, and
(0, 0, d/c) is the z-intercept (provided all of a, b, c, and d are
nonzero).  One way of regarding the "slope" of a plane is to write
down a unit vector which is perpendicular to it, called the normal
vector.  It is given by (a*I + b*J + c*K)/sqrt(a^2+b^2+c^2), where I, 
J, and K are the unit vectors in the x, y, and z directions.  The
coefficients of I, J, and K in this expression are called the
direction cosines of the vector, because they are the cosines of the
angles between the vector and the x-, y-, and z-axes, respectively.

A line is specified as the intersection of two nonparallel planes.
This means you need two linear equations in x, y, and z to determine
a line.  There are several standard forms for the equations of a line,
but a commonly used one is

  x - x0   y - y0   z - z0
  ------ = ------ = ------
    a        b        c

Here (x0, y0, z0) is a point on the line, and the numbers a, b, and c
determine the direction along the line: the vector a*I = b*J + c*K is
parallel to the line.  (Note: This form only works when the line is
not parallel to any of the xy-, xz-, or yz-planes, i.e., when neither 
a, b, or c is zero).

In some sense, the direction cosines are the closest analogue to the
slope.  In two dimensions, they are just the cosine of the 
inclination, which is the angle with the x-axis, and the cosine of its 
complement, which is the sine of the inclination. The slope is the 
ratio of these two, the tangent of the inclination. There is no exact 
analogue because there is no "ratio" of three direction cosines, or of 
any three numbers.

If you need further explanation, write to us again.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/