How do I determine whether a line is in a given plane in three-dimensional space?

1 Answer
Sep 17, 2015

Answer:

For plane:

#ax+bx+cx+d = 0#

and line:

#(x, y, z) = (x_0, y_0, z_0) + t(u, v, w)#

these are suitable conditions:

#{ (ax_0+by_0+cz_0+d = 0), (au+bv+cw = 0) :}#

Explanation:

The most general equation of a plane in three dimensional space is:

#ax+by+cz+d = 0#

A line in three dimensional space can be represented in a variety of ways, but one representation that will work for any line is the parametric form:

#(x, y, z) = (x_0, y_0, z_0) + t(u, v, w)#

This line will lie in the plane if and only if two distinct points of it both satisfy the equation of the plane.

Using #t = 0# and #t = 1#, we obtain the conditions:

#{ (ax_0+by_0+cz_0+d = 0), (a(x_0+u) + b(y_0 + v) + c(z_0 + w) + d = 0) :}#

Subtracting the first of these from the second, we obtain the condition:

#au+bv+cw = 0#

So a necessary and sufficient set of conditions is:

#{ (ax_0+by_0+cz_0+d = 0), (au+bv+cw = 0) :}#