Vectors in three dimensions
Problem 9.442 · easy
Express \( \displaystyle \overrightarrow{PQ} \) for \( \displaystyle P(6, 6, 5) \) and \( \displaystyle Q(-1, -2, 5) \) in component form, and find the unit vector in its direction.
- \[ \left[\begin{matrix}-7\\-8\\0\end{matrix}\right] \]Q − P.✓ Proved
- \[ \sqrt{113} \]‖PQ‖.✓ Proved
- \[ \left[\begin{matrix}- \frac{7 \sqrt{113}}{113}\\- \frac{8 \sqrt{113}}{113}\\0\end{matrix}\right] \]Divide by the length.✓ Proved
Answer \( \left\langle -7, -8, 0 \right\rangle,\ \text{unit vector}\ \left\langle - \frac{7 \sqrt{113}}{113}, - \frac{8 \sqrt{113}}{113}, 0 \right\rangle \)
✓ 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 |
| 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 vector components, magnitude, and unit vector. The steps are algebraically sound and match the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly computes the vector components, magnitude, and unit vector. The steps are algebraically sound and match the stated answer.qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly calculates the displacement vector Q - P and its magnitude, then normalizes it to find the unit vector. The algebraic steps are correct and the final answer matches the stated answer.gpt-oss:20b: pass 2026-10-09
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 2026-10-09 with SymPy 1.14.0.