Vectors in three dimensions
Problem 9.602 · easy
Express \( \displaystyle \overrightarrow{PQ} \) for \( \displaystyle P(-6, 0, 5) \) and \( \displaystyle Q(-1, 3, -6) \) in component form, and find the unit vector in its direction.
- \[ \left[\begin{matrix}5\\3\\-11\end{matrix}\right] \]Q − P.✓ Proved
- \[ \sqrt{155} \]‖PQ‖.✓ Proved
- \[ \left[\begin{matrix}\frac{\sqrt{155}}{31}\\\frac{3 \sqrt{155}}{155}\\- \frac{11 \sqrt{155}}{155}\end{matrix}\right] \]Divide by the length.✓ Proved
Answer \( \left\langle 5, 3, -11 \right\rangle,\ \text{unit vector}\ \left\langle \frac{\sqrt{155}}{31}, \frac{3 \sqrt{155}}{155}, - \frac{11 \sqrt{155}}{155} \right\rangle \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 |
| 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 computes the component vector and its magnitude, and the unit vector components are algebraically equivalent to the stated answer (e.g., sqrt(155)/31 = 5*sqrt(155)/155).
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly computes the component vector and its magnitude, and the unit vector components are algebraically equivalent to the stated answer (e.g., sqrt(155)/31 = 5*sqrt(155)/155).gpt-oss:20b: fail (error) 2026-10-13 — The unit vector is incorrectly computed. It should be (5/√155, 3/√155, −11/√155), not the fractions involving √155/31 and √155/155. The solution misapplies the division by the magnitude.qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly computes the vector components, magnitude, and unit vector. The algebraic simplifications are correct.
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_space, checked 2026-10-13 with SymPy 1.14.0.