# equation of the plane containing the points () and (1,3,5) and (9,8,4)

## Point 3: (x3,y3,z3)

Calculate the plane equation with points:
(,,) and (1,3,5) and (9,8,4)

## 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 1 3 5 1 8 4
 B =
 1 1 1 5 9 1 4
 C =
 1 1 3 1 9 8 1
 D =
 1 3 5 9 8 4

## Expand each determinant

|A| = y1(z2 - z3) + y2(z3 - z1) + y3(z1 - z2)
|A| =(5 - 4) + 3(4 - ) + 8( - 5)
|A| =(1) + 3(4) + 8(-5)
|A| =0 + 12 + -40
|A| =-28

|B| = z1(x2 - x3) + z2(x3 - x1) + z3(x1 - x2)
|B| = (1 - 9) + 5(9 - ) + 4( - 1)
|B| =(-8) + 5(9) + 4(-1)
|B| =0 + 45 + -4
|B| =41

|C| = x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)
|C| = (3 - 8) + 1(8 - ) + 9( - 3)
|C| =(-5) + 1(8) + 9(-3)
|C| =0 + 8 + -27
|C| =-19

|D| = x1(y2z3 - y3z2) + x2(y3z1 - y1z3) + x3(y1z2 - y2z1)
|D| = (3(4) - (8)5) + 1(8() - ()4) + 9((5) - (3))
|D| = (12 - 40) + 1(0 - 0) + 9(0 - 0)
|D| = (-28) + 1(0) + 9(0)
|D| = 0 + 0 + 0
|D| =0

## Build our equation of a plane:

-28x + 41y - 19z = 0

-28x + 41y - 19z = 0

### How does the Equation of a Plane Calculator work?

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?

1. 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)

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### 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