Vectors in three dimensions
Problem 9.527 · easy
Express \( \displaystyle \overrightarrow{PQ} \) for \( \displaystyle P(1, 0, -2) \) and \( \displaystyle Q(3, -4, -5) \) in component form, and find the unit vector in its direction.
- \[ \left[\begin{matrix}2\\-4\\-3\end{matrix}\right] \]Q − P.✓ Proved
- \[ \sqrt{29} \]‖PQ‖.✓ Proved
- \[ \left[\begin{matrix}\frac{2 \sqrt{29}}{29}\\- \frac{4 \sqrt{29}}{29}\\- \frac{3 \sqrt{29}}{29}\end{matrix}\right] \]Divide by the length.✓ Proved
Answer \( \left\langle 2, -4, -3 \right\rangle,\ \text{unit vector}\ \left\langle \frac{2 \sqrt{29}}{29}, - \frac{4 \sqrt{29}}{29}, - \frac{3 \sqrt{29}}{29} \right\rangle \)
✓ 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 and the unit vector. The algebraic steps are verified and the logic is sound.
Every verdict on record (4)
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 steps are verified and the logic is sound.qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly computes the vector components and the unit vector. The algebraic steps are verified, and the final answer matches the stated answer.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.