Equation of a plane
Problem 9.79 · hard
Find an equation of the plane through the points \( \displaystyle (-1, 2, 2),\ (-3, -1, 3),\ (2, -2, -2) \).
- \[ \left[\begin{matrix}-2\\-3\\1\end{matrix}\right] \]PQ.✓ Proved
- \[ \left[\begin{matrix}3\\-4\\-4\end{matrix}\right] \]PR.✓ Proved
- \[ \left[\begin{matrix}16\\-5\\17\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).✓ Proved
- \[ 8 \]n · P gives the constant.✓ Proved
- The plane is 16*x - 5*y + 17*z = 8.
Answer \( 16 x - 5 y + 17 z = 8 \)
Lines: 4 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | Not checked | — | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | all three points satisfy the equation |
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/plane_through_points, checked 2026-09-26 with SymPy 1.14.0.