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
