An equation of a plane represents a flat surface in three-dimensional space. A plane can be written in the form \(Ax+By+Cz=D\), where \(\langle A,B,C\rangle\) is a normal vector to the plane. These problems focus on finding the equation of a plane from three points using determinants, vectors, and coordinate patterns.
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.
\(\textbf{1)}\) \((1,2,3), (4,2,6), (-2,8,9)\)
The answer is \( 2x+3y-2z=2 \)
\(\,\,\,\,\,\, \left| {\begin{array}{ccc}x-x_1 & y-y_1 & z-z_1 \\x_2-x_1 & y_2-y_1 & z_2-z_1 \\x_3-x_1 & y_3-y_1 & z_3-z_1 \\\end{array} } \right|=0\)
\(\,\,\,\,\,\, \left| {\begin{array}{ccc}x-1 & y-2 & z-3 \\4-1 & 2-2 & 6-3 \\-2-1 & 8-2 & 9-3 \\\end{array} } \right|=0\)
\(\,\,\,\,\,\, \left| {\begin{array}{ccc}x-1 & y-2 & z-3 \\3 & 0 & 3 \\-3 & 6 & 6 \\\end{array} } \right|=0\)
\(\,\,\,\,\,\,(x-1)(-18) – (y-2)(27) + (z-3)(18) = 0\)
\(\,\,\,\,\,\,-18x – 27y + 18z = -18\)
\(\,\,\,\,\,\,\text{The answer is } 2x+3y-2z=2 \)
\(\textbf{2)}\) \((5,3,2), (7,7,7), (-1,0,5)\)
The answer is \( 3x-4y+2z=7 \)
\(\,\,\,\,\,\, \left| {\begin{array}{ccc}x-5 & y-3 & z-2 \\2 & 4 & 5 \\-6 & -3 & 3 \\\end{array} } \right|=0\)
\(\,\,\,\,\,\,(x-5)(27) + (y-3)(-36) + (z-2)(18) = 0\)
\(\,\,\,\,\,\,27x – 135 – 36y + 108 + 18z -36 = 0\)
\(\,\,\,\,\,\,27x-36y+18z-63 = 0\)
\(\,\,\,\,\,\,3x-4y+2z=7\)
\(\,\,\,\,\,\,\text{The answer is } 3x-4y+2z=7 \)
\(\textbf{3)}\) \((2,2,2), (4,-1,-1), (0,0,1)\)
The answer is \( 3x-8y+10z=10 \)
\(\,\,\,\,\,\, \left| {\begin{array}{ccc}x-2 & y-2 & z-2 \\2 & -3 & -3 \\-2 & -2 & -1 \\\end{array} } \right|=0\)
\(\,\,\,\,\,\,(x-2)(-3) – (y-2)(8) + (z-2)(-10) = 0\)
\(\,\,\,\,\,\, -3x+6 -8y +16 -10z+20 = 0\)
\(\,\,\,\,\,\, -3x-8y-10z+42=0\)
\(\,\,\,\,\,\,3x+8y+10z=42\)
\(\,\,\,\,\,\,\text{The answer is } 3x+8y+10z=42 \)
\(\textbf{4)}\) \((0,0,0), (1,1,1), (7,8,9)\)
The answer is \( x-2y+z=0 \)
\(\,\,\,\,\,\,\text{Use }(0,0,0)\text{ as the first point.}\)
\(\,\,\,\,\,\,\vec{v_1}=\langle1,1,1\rangle\)
\(\,\,\,\,\,\,\vec{v_2}=\langle7,8,9\rangle\)
\(\,\,\,\,\,\,\vec{n}=\vec{v_1}\times\vec{v_2}=\langle1,-2,1\rangle\)
\(\,\,\,\,\,\,1(x-0)-2(y-0)+1(z-0)=0\)
\(\,\,\,\,\,\,\text{The answer is }x-2y+z=0\)
\(\textbf{5)}\) \((1,2,3), (1,4,6), (1,8,-3)\)
The answer is \( x=1 \)
\(\,\,\,\,\,\,\text{Each point has }x=1.\)
\(\,\,\,\,\,\,(1,2,3)\text{ has }x=1\)
\(\,\,\,\,\,\,(1,4,6)\text{ has }x=1\)
\(\,\,\,\,\,\,(1,8,-3)\text{ has }x=1\)
\(\,\,\,\,\,\,\text{The plane through these points is }x=1\)
\(\,\,\,\,\,\,\text{The answer is }x=1\)
\(\textbf{6)}\) \((0,0,2), (1,0,2), (0,1,2)\)
The answer is \(z=2\)
\(\,\,\,\,\,\,\text{Each point has }z=2.\)
\(\,\,\,\,\,\,(0,0,2)\text{ has }z=2\)
\(\,\,\,\,\,\,(1,0,2)\text{ has }z=2\)
\(\,\,\,\,\,\,(0,1,2)\text{ has }z=2\)
\(\,\,\,\,\,\,\text{The plane through these points is }z=2\)
\(\,\,\,\,\,\,\text{The answer is }z=2\)
\(\textbf{7)}\) \((6,0,0), (0,6,0), (0,0,6)\)
The answer is \(x+y+z=6\)
\(\,\,\,\,\,\,\text{These are intercepts of the plane.}\)
\(\,\,\,\,\,\,\displaystyle\frac{x}{6}+\frac{y}{6}+\frac{z}{6}=1\)
\(\,\,\,\,\,\,x+y+z=6\)
\(\,\,\,\,\,\,\text{Check: }6+0+0=6\)
\(\,\,\,\,\,\,\text{Check: }0+6+0=6\)
\(\,\,\,\,\,\,\text{Check: }0+0+6=6\)
\(\,\,\,\,\,\,\text{The answer is }x+y+z=6\)
\(\textbf{8)}\) \((4,0,0), (0,2,0), (0,0,-4)\)
The answer is \(x+2y-z=4\)
\(\,\,\,\,\,\,\text{Use intercept form.}\)
\(\,\,\,\,\,\,\displaystyle\frac{x}{4}+\frac{y}{2}+\frac{z}{-4}=1\)
\(\,\,\,\,\,\,x+2y-z=4\)
\(\,\,\,\,\,\,\text{Check: }4+2(0)-0=4\)
\(\,\,\,\,\,\,\text{Check: }0+2(2)-0=4\)
\(\,\,\,\,\,\,\text{Check: }0+2(0)-(-4)=4\)
\(\,\,\,\,\,\,\text{The answer is }x+2y-z=4\)
\(\textbf{9)}\) \((2,0,1), (0,-1,2), (5,3,0)\)
The answer is \(2x-y+3z=7\)
\(\,\,\,\,\,\,\text{Use the form }Ax+By+Cz=D.\)
\(\,\,\,\,\,\,2(2)-0+3(1)=7\)
\(\,\,\,\,\,\,2(0)-(-1)+3(2)=7\)
\(\,\,\,\,\,\,2(5)-3+3(0)=7\)
\(\,\,\,\,\,\,\text{All three points satisfy }2x-y+3z=7.\)
\(\,\,\,\,\,\,\text{The answer is }2x-y+3z=7\)
\(\textbf{10)}\) \((0,3,0), (2,3,5), (-1,3,4)\)
The answer is \(y=3\)
\(\,\,\,\,\,\,\text{Each point has }y=3.\)
\(\,\,\,\,\,\,(0,3,0)\text{ has }y=3\)
\(\,\,\,\,\,\,(2,3,5)\text{ has }y=3\)
\(\,\,\,\,\,\,(-1,3,4)\text{ has }y=3\)
\(\,\,\,\,\,\,\text{The plane through these points is }y=3\)
\(\,\,\,\,\,\,\text{The answer is }y=3\)
\(\textbf{11)}\) \((0,0,0), (2,1,0), (1,1,1)\)
The answer is \(x-2y+z=0\)
\(\,\,\,\,\,\,\text{Use }(0,0,0)\text{ as the first point.}\)
\(\,\,\,\,\,\,\vec{v_1}=\langle2,1,0\rangle\)
\(\,\,\,\,\,\,\vec{v_2}=\langle1,1,1\rangle\)
\(\,\,\,\,\,\,\vec{n}=\vec{v_1}\times\vec{v_2}=\langle1,-2,1\rangle\)
\(\,\,\,\,\,\,1(x-0)-2(y-0)+1(z-0)=0\)
\(\,\,\,\,\,\,\text{The answer is }x-2y+z=0\)
\(\textbf{12)}\) \((2,0,1), (1,2,0), (0,5,0)\)
The answer is \(3x+y-z=5\)
\(\,\,\,\,\,\,\text{Use the form }Ax+By+Cz=D.\)
\(\,\,\,\,\,\,3(2)+0-1=5\)
\(\,\,\,\,\,\,3(1)+2-0=5\)
\(\,\,\,\,\,\,3(0)+5-0=5\)
\(\,\,\,\,\,\,\text{All three points satisfy }3x+y-z=5.\)
\(\,\,\,\,\,\,\text{The answer is }3x+y-z=5\)
\(\textbf{13)}\) \((4,0,0), (0,4,1), (1,3,-2)\)
The answer is \(x+y=4\)
\(\,\,\,\,\,\,\text{Notice that each point has }x+y=4.\)
\(\,\,\,\,\,\,4+0=4\)
\(\,\,\,\,\,\,0+4=4\)
\(\,\,\,\,\,\,1+3=4\)
\(\,\,\,\,\,\,\text{The equation does not depend on }z.\)
\(\,\,\,\,\,\,\text{The answer is }x+y=4\)
\(\textbf{14)}\) \((0,0,0), (1,0,1), (0,2,2)\)
The answer is \(x+y-z=0\)
\(\,\,\,\,\,\,\text{Use }(0,0,0)\text{ as the first point.}\)
\(\,\,\,\,\,\,\vec{v_1}=\langle1,0,1\rangle\)
\(\,\,\,\,\,\,\vec{v_2}=\langle0,2,2\rangle\)
\(\,\,\,\,\,\,\vec{n}=\vec{v_1}\times\vec{v_2}=\langle-2,-2,2\rangle\)
\(\,\,\,\,\,\,\text{Simplify the normal vector to }\langle1,1,-1\rangle.\)
\(\,\,\,\,\,\,x+y-z=0\)
\(\,\,\,\,\,\,\text{The answer is }x+y-z=0\)
\(\textbf{15)}\) \((5,0,0), (2,2,0), (1,1,5)\)
The answer is \(2x+3y+z=10\)
\(\,\,\,\,\,\,\text{Use the form }Ax+By+Cz=D.\)
\(\,\,\,\,\,\,2(5)+3(0)+0=10\)
\(\,\,\,\,\,\,2(2)+3(2)+0=10\)
\(\,\,\,\,\,\,2(1)+3(1)+5=10\)
\(\,\,\,\,\,\,\text{All three points satisfy }2x+3y+z=10.\)
\(\,\,\,\,\,\,\text{The answer is }2x+3y+z=10\)
\(\textbf{16)}\) \((2,0,0), (5,1,1), (-1,2,-1)\)
The answer is \(x-3z=2\)
\(\,\,\,\,\,\,\text{Notice that each point has }x-3z=2.\)
\(\,\,\,\,\,\,2-3(0)=2\)
\(\,\,\,\,\,\,5-3(1)=2\)
\(\,\,\,\,\,\,-1-3(-1)=2\)
\(\,\,\,\,\,\,\text{The equation does not depend on }y.\)
\(\,\,\,\,\,\,\text{The answer is }x-3z=2\)
\(\textbf{17)}\) \((1,2,0), (-1,0,0), (3,0,2)\)
The answer is \(x-y-2z=-1\)
\(\,\,\,\,\,\,\text{Use the form }Ax+By+Cz=D.\)
\(\,\,\,\,\,\,1-2-2(0)=-1\)
\(\,\,\,\,\,\,-1-0-2(0)=-1\)
\(\,\,\,\,\,\,3-0-2(2)=-1\)
\(\,\,\,\,\,\,\text{All three points satisfy }x-y-2z=-1.\)
\(\,\,\,\,\,\,\text{The answer is }x-y-2z=-1\)
\(\textbf{18)}\) \((3,0,0), (0,2,3), (1,-1,2)\)
The answer is \(x+z=3\)
\(\,\,\,\,\,\,\text{Notice that each point has }x+z=3.\)
\(\,\,\,\,\,\,3+0=3\)
\(\,\,\,\,\,\,0+3=3\)
\(\,\,\,\,\,\,1+2=3\)
\(\,\,\,\,\,\,\text{The equation does not depend on }y.\)
\(\,\,\,\,\,\,\text{The answer is }x+z=3\)
\(\textbf{19)}\) \((0,2,0), (1,6,0), (0,0,-2)\)
The answer is \(4x-y+z=-2\)
\(\,\,\,\,\,\,\text{Use the form }Ax+By+Cz=D.\)
\(\,\,\,\,\,\,4(0)-2+0=-2\)
\(\,\,\,\,\,\,4(1)-6+0=-2\)
\(\,\,\,\,\,\,4(0)-0+(-2)=-2\)
\(\,\,\,\,\,\,\text{All three points satisfy }4x-y+z=-2.\)
\(\,\,\,\,\,\,\text{The answer is }4x-y+z=-2\)
\(\textbf{20)}\) \((12,0,0), (0,6,0), (0,0,4)\)
The answer is \(x+2y+3z=12\)
\(\,\,\,\,\,\,\text{Use intercept form.}\)
\(\,\,\,\,\,\,\displaystyle\frac{x}{12}+\frac{y}{6}+\frac{z}{4}=1\)
\(\,\,\,\,\,\,x+2y+3z=12\)
\(\,\,\,\,\,\,\text{Check: }12+2(0)+3(0)=12\)
\(\,\,\,\,\,\,\text{Check: }0+2(6)+3(0)=12\)
\(\,\,\,\,\,\,\text{Check: }0+2(0)+3(4)=12\)
\(\,\,\,\,\,\,\text{The answer is }x+2y+3z=12\)
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