Vectors in three dimensions
Problem 9.603 · easy
Express \( \displaystyle \overrightarrow{PQ} \) for \( \displaystyle P(2, 4, 2) \) and \( \displaystyle Q(-2, -1, 5) \) in component form, and find the unit vector in its direction.
- \[ \left[\begin{matrix}-4\\-5\\3\end{matrix}\right] \]Q − P.✓ Proved
- \[ 5 \sqrt{2} \]‖PQ‖.✓ Proved
- \[ \left[\begin{matrix}- \frac{2 \sqrt{2}}{5}\\\frac{\left(-1\right) \sqrt{2}}{2}\\\frac{3 \sqrt{2}}{10}\end{matrix}\right] = \left[\begin{matrix}- \frac{2 \sqrt{2}}{5}\\- \frac{\sqrt{2}}{2}\\\frac{3 \sqrt{2}}{10}\end{matrix}\right] \]Divide by the length.✓ Proved
Answer \( \left\langle -4, -5, 3 \right\rangle,\ \text{unit vector}\ \left\langle - \frac{2 \sqrt{2}}{5}, - \frac{\sqrt{2}}{2}, \frac{3 \sqrt{2}}{10} \right\rangle \)
✓ 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, the magnitude, and the unit vector. The algebraic steps are verified and the logic is sound.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly computes the vector components, the magnitude, and the unit vector. The algebraic steps are verified and the logic is sound.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly computes the vector components and the unit vector. The algebraic steps are verified, and the setup matches the problem statement.
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.