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(xp,yp,zp), Q(xq,yq,zq) and R(xr,yr,zr).
We first define vector →n as the cross product of vectors →PR and →PQ
From definition of the cross product, →n is perpendicular to both vectors →PR and →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 →n=<xn,yn,zn> and vectors →PM=<x−xp,y−yp,z−zp> is equal to zero.
QUESTION: Find the equation of plane through the points \( P(1,-2,-5) \) , \( Q(1,2,3) \) and \( R(0,3,4) \)
STEP 1: Find vectors PR and PQ and calculate vector n = PR x PQ.
\( \vec{PR} = \lt x_r - x_p , y_r - y_p , z_r - z_p \gt = \lt 0 - (1) , 3 - (-2), 4 - (-5) \gt = \lt -1, 5 , 9 \gt \)
\( \vec{PQ} = \lt x_q - x_p , y_q - y_p , z_q - z_p \gt = \vec{PQ} = \lt 1 - (1) , 2 - (-2), 3 - (-5) \gt = \lt 0, 4 , 8 \gt \)
\( \vec {n} = \vec {PR} \times \vec {PQ} = \begin{vmatrix}\vec i & \vec j & \vec k \\-1 & 5 & 9\\0 & 4 & 8\end{vmatrix} = \lt 4,8,-4\gt \)
\( \vec{PR} = \lt x_r - x_p , y_r - y_p , z_r - z_p \gt = \lt 0 - (1) , 3 - (-2), 4 - (-5) \gt = \lt -1, 5 , 9 \gt \)
\( \vec{PQ} = \lt x_q - x_p , y_q - y_p , z_q - z_p \gt = \vec{PQ} = \lt 1 - (1) , 2 - (-2), 3 - (-5) \gt = \lt 0, 4 , 8 \gt \)
\( \vec {n} = \vec {PR} \times \vec {PQ} = \begin{vmatrix}\vec i & \vec j & \vec k \\-1 & 5 & 9\\0 & 4 & 8\end{vmatrix} = \lt 4,8,-4\gt \)
STEP 2: Write the components of vector PM.
\( \vec{PM} = \lt x - (1), y - (-2),z - (-5) \gt = \lt x -1, y+2,z+5 \gt \)
\( \vec{PM} = \lt x - (1), y - (-2),z - (-5) \gt = \lt x -1, y+2,z+5 \gt \)
STEP 3: Write that the dot product of vectors n and PM is equal to zero.
\( \vec n \cdot \vec {PM} = \lt 4 , 8, -4, \gt \cdot \lt x -1, y+2,z+5 \gt = 0 \)
which gives
\( 4 \cdot (x-1) +8 \cdot (y+2) -4 \cdot (z+5) = 0 \)
\( \vec n \cdot \vec {PM} = \lt 4 , 8, -4, \gt \cdot \lt x -1, y+2,z+5 \gt = 0 \)
which gives
\( 4 \cdot (x-1) +8 \cdot (y+2) -4 \cdot (z+5) = 0 \)
STEP 4: Expand the product, simplify and write the equation in the form a x + b y + c z = d.
\(4x+8y-4z = 8\)
\(4x+8y-4z = 8\)