Vectors in the plane
Problem 9.597 · easy
Find \( \displaystyle \overrightarrow{PQ} \) in component form and its length, for \( \displaystyle P(1, -6) \) and \( \displaystyle Q(6, 1) \).
- \[ \left[\begin{matrix}5\\7\end{matrix}\right] \]Terminal point minus initial point.✓ Proved
- \[ \sqrt{74} \]√(x² + y²).✓ Proved
Answer \( \left\langle 5, 7 \right\rangle,\ \|\overrightarrow{PQ}\| = \sqrt{74} \)
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the component form as the difference between terminal and initial points and applies the distance formula correctly.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the component form as the difference between terminal and initial points and applies the distance formula correctly.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly calculates the component vector by subtracting the coordinates of P from Q and computes the magnitude using the standard distance formula. The results match the stated answer.
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/vectors_plane, checked 2026-10-13 with SymPy 1.14.0.