Vectors in three dimensions
Problem 9.530 · easy
Express \( \displaystyle \overrightarrow{PQ} \) for \( \displaystyle P(-3, -3, 2) \) and \( \displaystyle Q(4, 5, 3) \) in component form, and find the unit vector in its direction.
- \[ \left[\begin{matrix}7\\8\\1\end{matrix}\right] \]Q − P.✓ Proved
- \[ \sqrt{114} \]‖PQ‖.✓ Proved
- \[ \left[\begin{matrix}\frac{7 \sqrt{114}}{114}\\\frac{4 \sqrt{114}}{57}\\\frac{\sqrt{114}}{114}\end{matrix}\right] \]Divide by the length.✓ Proved
Answer \( \left\langle 7, 8, 1 \right\rangle,\ \text{unit vector}\ \left\langle \frac{7 \sqrt{114}}{114}, \frac{4 \sqrt{114}}{57}, \frac{\sqrt{114}}{114} \right\rangle \)
✓ 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 algebraic steps are verified and the final answer matches the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly computes the vector components, magnitude, and unit vector. The algebraic steps are verified and the final answer matches the stated answer.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly computes the vector components and the unit vector. The algebraic simplifications are correct (e.g., 8/sqrt(114) = 4*sqrt(114)/57).gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.