Vectors in three dimensions
Problem 9.368 · medium
Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 8 \) pointing in the same direction as \( \displaystyle \mathbf v = \left\langle 2, 1, 2 \right\rangle \).
- \[ 3 \]‖v‖.✓ Proved
- \[ \left[\begin{matrix}\frac{16}{3}\\\frac{8}{3}\\\frac{16}{3}\end{matrix}\right] \]8·v/‖v‖.✓ Proved
Answer \( \left\langle \frac{16}{3}, \frac{8}{3}, \frac{16}{3} \right\rangle \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 checked, and parallel by the equality case of Cauchy–Schwarz |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly calculates the magnitude of v as 3 and scales v by 8/3 to obtain the desired vector. The algebraic steps are verified and the logic is sound.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly calculates the magnitude of v as 3 and scales v by 8/3 to obtain the desired vector. The algebraic steps are verified and the logic is sound.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution claims the norm of v is 3, but the actual norm is sqrt(2^2 + 1^2 + 2^2) = 3. Wait, sqrt(9) is 3. So the norm calculation is correct. Let me re-read the prompt constraints. 'Each line of the solution is either an EQUATION... or a SENTENCE'. The provided solution only contains equations. There are no sentences to review for logical flow or theorem application. However, the prompt asks me to review 'worked calculus solutions'. If the solution is just equations, I must check if the equations are correct and if they constitute a valid solution. Equation 1 says 3=3, labeled as ||v||. This is correct. Equation 2 says the result vector equals 8*v/||v||. This is correct. The final answer is correct. Why would it fail? Ah, the prompt says 'Review it.' and 'Each line... is either an EQUATION... or a SENTENCE'. The input provided is just two equations. Is there a missing step? No, the logic is sound. But wait, the prompt says 'Your job is to find what is wrong'. If everything is right, I should say pass. Let me double check the norm. v = <2, 1, 2>. ||v|| = sqrt(4+1+4) = sqrt(9) = 3. Correct. u = 8 * <2, 1, 2> / 3 = <16/3, 8/3, 16/3>. Correct. The solution is correct. I will mark it as pass.gpt-oss:20b: pass 2026-10-07
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 2026-10-07 with SymPy 1.14.0.