Notes

 

\(\text{Equation of a Plane Given 3 Points}\)
\( \left| {\begin{array}{ccc}x-x_1 & y-y_1 & z-z_1 \\x_2-x_1 & y_2-y_1 & z_2-z_1 \\x_3-x_1 & y_3-y_1 & z_3-z_1 \\\end{array} } \right|=0\)

 

 

Practice Problems

Find the equation of the plane through the following points.

\(\textbf{1)}\) \((1,2,3), (4,2,6), (-2,8,9)\)

Show Equation of the Plane

The answer is \( 2x+3y-2z=2 \)

Show Work

\(\,\,\,\,\,\, \left| {\begin{array}{ccc}x-x_1 & y-y_1 & z-z_1 \\x_2-x_1 & y_2-y_1 & z_2-z_1 \\x_3-x_1 & y_3-y_1 & z_3-z_1 \\\end{array} } \right|=0\)
\(\,\,\,\,\,\, \left| {\begin{array}{ccc}x-1 & y-2 & z-3 \\4-1 & 2-2 & 6-3 \\-2-1 & 8-2 & 9-3 \\\end{array} } \right|=0\)
\(\,\,\,\,\,\, \left| {\begin{array}{ccc}x-1 & y-2 & z-3 \\3 & 0 & 3 \\-3 & 6 & 6 \\\end{array} } \right|=0\)
\(\,\,\,\,\,\,(x-1)(0 \cdot 6-3 \cdot 6) – (y-2)(3 \cdot 6-3 \cdot (-3)) + (z-3)(3 \cdot 6-0 \cdot (-3)) = 0\)
\(\,\,\,\,\,\,(x-1)(-18) – (y-2)(27) + (z-3)(18) = 0\)
\(\,\,\,\,\,\,-18x + 18 – 27y + 54 + 18z -54 = 0\)
\(\,\,\,\,\,\,-18x – 27y + 18z = -18\)
\(\,\,\,\,\,\,\text{The answer is } 2x+3y-2z=2 \)

 

\(\textbf{2)}\) \((5,3,2), (7,7,7), (-1,0,5)\)

Show Equation of the Plane

The answer is \( 3x-4y+2z=7 \)

Show Work

\(\,\,\,\,\,\, \left| {\begin{array}{ccc}x-x_1 & y-y_1 & z-z_1 \\x_2-x_1 & y_2-y_1 & z_2-z_1 \\x_3-x_1 & y_3-y_1 & z_3-z_1 \\\end{array} } \right|=0\)
\(\,\,\,\,\,\, \left| {\begin{array}{ccc}x-5 & y-3 & z-2 \\7-5 & 7-3 & 7-2 \\-1-5 & 0-3 & 5-2 \\\end{array} } \right|=0\)
\(\,\,\,\,\,\, \left| {\begin{array}{ccc}x-5 & y-3 & z-2 \\2 & 4 & 5 \\-6 & -3 & 3 \\\end{array} } \right|=0\)
\(\,\,\,\,\,\,(x-5)(4 \cdot 3-5 \cdot (-3)) – (y-3)(2 \cdot 3-5 \cdot (-6)) + (z-2)( 2 \cdot (-3)-4 \cdot (-6)) = 0\)
\(\,\,\,\,\,\,(x-5)(27) + (y-3)(-36) + (z-2)(18) = 0\)
\(\,\,\,\,\,\,27x – 135 – 36y + 108 + 18z -36 = 0\)
\(\,\,\,\,\,\,27x-36y+18z-63 = 0\)
\(\,\,\,\,\,\,\text{The answer is } 3x-4y+2z=7 \)

 

\(\textbf{3)}\) \((2,2,2), (4,-1,-1), (0,0,1)\)

Show Equation of the Plane

The answer is \( 3x-8y+10z=10 \)

Show Work

\(\,\,\,\,\,\, \left| {\begin{array}{ccc}x-x_1 & y-y_1 & z-z_1 \\x_2-x_1 & y_2-y_1 & z_2-z_1 \\x_3-x_1 & y_3-y_1 & z_3-z_1 \\\end{array} } \right|=0\)
\(\,\,\,\,\,\, \left| {\begin{array}{ccc}x-2 & y-2 & z-2 \\4-2 & -1-2 & -1-2 \\0-2 & 0-2 & 1-2 \\\end{array} } \right|=0\)
\(\,\,\,\,\,\, \left| {\begin{array}{ccc}x-2 & y-2 & z-2 \\2 & -3 & -3 \\-2 & -2 & -1 \\\end{array} } \right|=0\)
\(\,\,\,\,\,\,(x-2)(-3 \cdot (-1)-(-3) \cdot (-2)) – (y-2)(2 \cdot (-1)-(-3) \cdot (-2)) + (z-2)(2 \cdot (-2)-(-3) \cdot (-2)) = 0\)
\(\,\,\,\,\,\,(x-2)(-3) – (y-2)(8) + (z-2)(-10) = 0\)
\(\,\,\,\,\,\, -3x+6 -8y +16 -10z+20 = 0\)
\(\,\,\,\,\,\,\text{The answer is } 3x-8y+10z=10 \)

 

\(\textbf{4)}\) \((0,0,0), (1,1,1), (7,8,9)\)

Show Equation of the Plane

The answer is \( x-2y+z=0 \)

 

\(\textbf{5)}\) \((1,2,3), (1,4,6), (1,8,-3)\)

Show Equation of the Plane

The answer is \( x=1 \)

 

See Related Pages\(\)

\(\bullet\text{ Displacement Vectors}\)
\(\,\,\,\,\,\,\,\,(x_2-x_1)\vec{i}+(y_2-y_1)\vec{j}…\)

\(\bullet\text{ Magnitude, Direction, and Unit Vectors}\)
\(\,\,\,\,\,\,\,\,|\vec{u}|=\sqrt{a^2+b^2}…\)

\(\bullet\text{ Dot Product}\)
\(\,\,\,\,\,\,\,\,a \cdot b=x_1 x_2+ y_1 y_2…\)

\(\bullet\text{ Parallel and Perpendicular Vectors}\)
\(\,\,\,\,\,\,\,\,⟨8,2⟩ \text{ and } ⟨−4,−1⟩…\)

\(\bullet\text{ Scalar and Vector Projections}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{a \cdot b}{|b|^2} \, \vec{b}…\)

\(\bullet\text{ Cross Product}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Equation of a Plane}\)
\(\,\,\,\,\,\,\,\,Ax+By+Cz=D…\)

\(\bullet\text{ Andymath Homepage}\)

 

In Summary

An equation of a plane is a mathematical representation of a flat surface that extends indefinitely in all directions. A plane can be defined by 3 non-collinear points.