How do I determine whether a line is in a given plane in three-dimensional space?
1 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
#{ (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) :}#
