In this section, we’ll discuss how to write the equation for a straight line in coordinate form. There are essentially two different ways of doing so:
UNSYMMETRICAL FORM OF THE EQUATION OF A LINE
A line can be defined as the intersection of two planes. Thus, the equations of two planes considered together represents a straight line. For example, the set of equations
represents the straight line formed by the intersection of these two planes.
Recall that the planes will intersect only if they are non-parallel, i.e., only if
SYMMETRICAL FORM OF THE EQUATION OF A LINE:
Consider a line with direction cosines , , and passing through the point For any point on this line, the set of numbers must be proportional to the direction cosines, as has already been discussed. Thus, the equation of this line can be written as
Extending this, we can write the equation of the line passing through and as
Note that for any point at a distance from along the line with direction cosines , , , we have
Thus, the coordinates of can be written as
This is a useful fact and we’ll be using it frequently.
Find the direction cosines of the line
We have ,
Comparing this with the symmetrical form of the equation of a line, we can say that the direction ratios of this line are proportional to , , . Thus, the direction cosines are