Equation of a Plane Through three Points
An interactive worksheet including a calculator and solver to find the equation of a plane through three points is presented. As many examples as needed may be generated interactively along with their solutions and explanations. Below is shown a plane passing through the three points \( P(x_p,y_p,z_p) \), \( Q(x_q,y_q,z_q) \) and \( R(x_r,y_r,z_r) \).
We first define vector \( \vec {n} \) as the cross product of vectors \( \vec {PR} \) and \( \vec {PQ} \)
From definition of the cross product, \( \vec {n} \) is perpendicular to both vectors \( \vec {PR} \) and \( \vec {PQ} \) and therefore to the plane containing all three points P, Q and R. Any point \( M(x,y,z) \) is on the plane if the dot product of \( \vec n = \lt x_n,y_n,z_n \gt \) and vectors \( \vec {PM} = \lt x  x_p , y  y_p , z  z_p \gt \) is equal to zero.
STEP 1: Find vectors PR and PQ and calculate vector n = PR x PQ.
STEP 2: Write the components of vector PM.
STEP 3: Write that the dot product of vectors n and PM is equal to zero.
STEP 4: Expand the product, simplify and write the equation in the form a x + b y + c z = d.
