Vectors in the plane
Problem 9.512 · easy
Find the unit vector in the direction of \( \displaystyle \mathbf v = \left\langle -12, -16 \right\rangle \).
- \[ 20 \]‖v‖.✓ Proved
- \[ \left[\begin{matrix}- \frac{3}{5}\\- \frac{4}{5}\end{matrix}\right] \]v/‖v‖.✓ Proved
Answer \( \left\langle - \frac{3}{5}, - \frac{4}{5} \right\rangle \)
✓ 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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | length 1 and parallel to v, same direction |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly calculates the magnitude and normalizes the vector. The lack of explanatory text is not a logical error.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly calculates the magnitude and normalizes the vector. The lack of explanatory text is not a logical error.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly calculates the magnitude of the vector and divides the vector by this magnitude to find the unit vector. The algebraic steps are correct.
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_plane, checked 2026-10-11 with SymPy 1.14.0.