Equation of a Plane - 3 Points
Main Concept
A plane can be defined by four different methods:
A line and a point not on the line
Three non-collinear points (three points not on a line)
A point and a normal vector
Two intersecting lines
Two parallel and non-coincident lines
The Cartesian equation of a plane is a⋅x+ b⋅y + c⋅z + d = 0, where a,b,c is the vector normal to the plane.
How to find the equation of a plane using three non-collinear points
Three points (A,B,C) can define two distinct vectors AB and AC. Since the two vectors lie on the plane, their cross product can be used as a normal to the plane.
Determine the vectors
Find the cross product of the two vectors
Substitute one point into the Cartesian equation to solve for d.
Example
Find the equation of the plane that passes through the points A = 1,1,1, B =−1,1,0, C = 2,0,3.
AB⇀ =
xB− xAi⏞ +yB − yAj⏞ +zB − zAk⏞
−1−1i⏞ +1−1j⏞ +0−1k⏞
−2⋅i⏞ −k⏞
AC⇀ =
xC− xAi⏞ +yC − yAj⏞ +zC − zAk⏞
2−1i⏞ +0−1j⏞ +3−1k⏞
i⏞ −j⏞ + 2⋅k⏞
Determine the normal vector
AB⇀ x AC⇀ = i⏞j⏞k⏞−20−11−12
AB⇀ x AC⇀ = −i⏞+3⋅j⏞ +2⋅k⏞
The equation of the plane is
−x + 3⋅y + 2⋅ z + d = 0
Plug in any point to find the value of d
−d=
−x + 3⋅y + 2⋅ z
−1 + 3⋅ 1 + 2⋅ 1
d=
−4
The equation of the plane is −x + 3⋅y + 2⋅z −4 =0
Change the three points on the plane and see how it affects the plane.
Point A
Point B
xA =
xB=
yA=
yB=
zA=
zB=
Point C
xC =
yC =
zC =
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