Equation of a plane
Problem 9.43 · hard
Find an equation of the plane through the points \( \displaystyle (2, 0, -1),\ (-2, 1, 2),\ (3, -2, -2) \).
- \[ \left[\begin{matrix}-4\\1\\3\end{matrix}\right] \]PQ.✓ Proved
- \[ \left[\begin{matrix}1\\-2\\-1\end{matrix}\right] \]PR.✓ Proved
- \[ \left[\begin{matrix}5\\-1\\7\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).✓ Proved
- \[ 3 \]n · P gives the constant.✓ Proved
- The plane is 5*x - y + 7*z = 3.
Answer \( 5 x - y + 7 z = 3 \)
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.