Vectors in three dimensions
Problem 9.287 · easy
Express \( \displaystyle \overrightarrow{PQ} \) for \( \displaystyle P(0, 5, 5) \) and \( \displaystyle Q(-4, 0, -3) \) in component form, and find the unit vector in its direction.
- \[ \left[\begin{matrix}-4\\-5\\-8\end{matrix}\right] \]Q − P.✓ Proved
- \[ \sqrt{105} \]‖PQ‖.✓ Proved
- \[ \left[\begin{matrix}- \frac{4 \sqrt{105}}{105}\\\frac{\left(-1\right) \sqrt{105}}{21}\\- \frac{8 \sqrt{105}}{105}\end{matrix}\right] = \left[\begin{matrix}- \frac{4 \sqrt{105}}{105}\\- \frac{\sqrt{105}}{21}\\- \frac{8 \sqrt{105}}{105}\end{matrix}\right] \]Divide by the length.✓ Proved
Answer \( \left\langle -4, -5, -8 \right\rangle,\ \text{unit vector}\ \left\langle - \frac{4 \sqrt{105}}{105}, - \frac{\sqrt{105}}{21}, - \frac{8 \sqrt{105}}{105} \right\rangle \)
✓ 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 logically sound and the algebra is verified.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly computes the vector components, magnitude, and unit vector. The steps are logically sound and the algebra is verified.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly computes the vector components and the unit vector. The algebraic simplifications are correct.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.