Point 1: (x1,y1,z1)


Point 2: (x2,y2,z2)


Point 3: (x3,y3,z3)




    Take the Quiz

Calculate the plane equation with points:

(4,2,3) and (4,7,6) and (7,5,9)

Standard equation for a plane

Ax + By + Cz + D = 0

Calculate Determinants

  
 A = 
  
 1  y1  z1 
 1  y2  z2 
 1  y3  z3 

  
 B = 
  
 x1  1  z1 
 x2  1  z2 
 x3  1  z3 

  
 C = 
  
 x1  y1  1 
 x2  y2  1 
 x3  y3  1 

  
 D = 
  
 x1  y1  z1 
 x2  y2  z2 
 x3  y3  z3 

Plug in our point values

  
 A = 
  
 1  2  3 
 1  7  6 
 1  5  9 

  
 B = 
  
 4  1  3 
 4  1  6 
 7  1  9 

  
 C = 
  
 4  2  1 
 4  7  1 
 7  5  1 

  
 D = 
  
 4  2  3 
 4  7  6 
 7  5  9 

Expand the determinant |A|

|A| = y1(z2 - z3) + y2(z3 - z1) + y3(z1 - z2)

|A| =2(6 - 9) + 7(9 - 3) + 5(3 - 6)

|A| =2(-3) + 7(6) + 5(-3)

|A| =-6 + 42 + -15

|A| =21

Expand the determinant |B|

|B| = z1(x2 - x3) + z2(x3 - x1) + z3(x1 - x2)

|B| = 3(4 - 7) + 6(7 - 4) + 9(4 - 4)

|B| =3(-3) + 6(3) + 9(0)

|B| =-9 + 18 + 0

|B| =9

Expand the determinant |C|

|C| = x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)

|C| = 4(7 - 5) + 4(5 - 2) + 7(2 - 7)

|C| =4(2) + 4(3) + 7(-5)

|C| =8 + 12 + -35

|C| =-15

Expand the determinant |D|

|D| = x1(y2z3 - y3z2) + x2(y3z1 - y1z3) + x3(y1z2 - y2z1)

|D| = 4(7(9) - (5)6) + 4(5(3) - (2)9) + 7(2(6) - (7)3)

|D| = 4(63 - 30) + 4(15 - 18) + 7(12 - 21)

|D| = 4(33) + 4(-3) + 7(-9)

|D| = 132 + -12 + -63

|D| =57

Final Answer

21x + 9y - 15z + 57 = 0


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What is the Answer?
21x + 9y - 15z + 57 = 0
How does the Equation of a Plane Calculator work?
Free Equation of a Plane Calculator - Given three 3-dimensional points, this calculates the equation of a plane that contains those points.
This calculator has 3 inputs.

What 1 formula is used for the Equation of a Plane Calculator?

Ax + By + Cz + D = 0
|A| = y1(z2 - z3) + y2(z3 - z1) + y3(z1 - z2)
|B| = z1(x2 - x3) + z2(x3 - x1) + z3(x1 - x2)
|C| = x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)
|D| = x1(y2z3 - y3z2) + x2(y3z1 - y1z3) + x3(y1z2 - y2z1)


For more math formulas, check out our Formula Dossier

What 6 concepts are covered in the Equation of a Plane Calculator?

determinant
value computed from a square matrix
det(A) or |A|
equation
a statement declaring two mathematical expressions are equal
equation of a plane
formula to graph the points in a plane
Ax + By + Cz + D = 0
matrix
a rectangular array of numbers or symbols which are generally arranged in rows and columns
plane
a flat, two-dimensional surface that extends indefinitely
point
an exact location in the space, and has no length, width, or thickness
Example calculations for the Equation of a Plane Calculator
Equation of a Plane Calculator Video

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