Home›Calculus 3›Projections and orthogonality›Problem 9.269

Projections and orthogonality

Problem 9.269 · medium

Find the scalar projection \( \displaystyle \operatorname{comp}_{\mathbf v} \mathbf u \) for \( \displaystyle \mathbf u = \left\langle 4, 0, -2 \right\rangle \), \( \displaystyle \mathbf v = \left\langle 0, -5, -5 \right\rangle \).

\[ 0 + 0 + 10 = 10 \]
u·v.✓ Proved
\[ 0 + 25 + 25 = 50 \]
‖v‖².✓ Proved
\[ \sqrt{2} \]
comp = u·v / ‖v‖.✓ Proved

Answer \( \sqrt{2} \)

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
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	‖u‖ cos θ with θ from arccos

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution uses the formula for scalar projection (u·v / ||v||) but fails to compute ||v|| correctly from the given ||v||² = 50. ||v|| should be sqrt(50) = 5*sqrt(2), making the correct projection 10 / (5*sqrt(2)) = sqrt(2). While the final numerical answer happens to be correct, the intermediate step 'sqrt(2) = sqrt(2)' is nonsensical and does not show the division by the norm. More critically, the label 'comp = u·v / ||v||' implies the calculation was done, but the equation shown is an identity, not the calculation. A student would not learn how to perform the division. However, looking closer, the final answer IS correct. Is it an error? The line 'sqrt(2) = sqrt(2)' is presented as the result of 'comp = u·v / ||v||'. This is mathematically false as a derivation step because 10 / sqrt(50) is not trivially sqrt(2) without simplification. It skips the crucial step of calculating the norm and dividing. It is misleading because it presents a tautology as the calculation result.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution uses the formula for scalar projection (u·v / ||v||) but fails to compute ||v|| correctly from the given ||v||² = 50. ||v|| should be sqrt(50) = 5*sqrt(2), making the correct projection 10 / (5*sqrt(2)) = sqrt(2). While the final numerical answer happens to be correct, the intermediate step 'sqrt(2) = sqrt(2)' is nonsensical and does not show the division by the norm. More critically, the label 'comp = u·v / ||v||' implies the calculation was done, but the equation shown is an identity, not the calculation. A student would not learn how to perform the division. However, looking closer, the final answer IS correct. Is it an error? The line 'sqrt(2) = sqrt(2)' is presented as the result of 'comp = u·v / ||v||'. This is mathematically false as a derivation step because 10 / sqrt(50) is not trivially sqrt(2) without simplification. It skips the crucial step of calculating the norm and dividing. It is misleading because it presents a tautology as the calculation result.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution uses the formula for scalar projection, comp_v u = (u · v) / ||v||, but fails to compute ||v|| correctly. Line 2 computes ||v||^2 = 50, so ||v|| = sqrt(50) = 5*sqrt(2). The correct projection is 10 / (5*sqrt(2)) = sqrt(2). While the final answer is numerically correct, Line 3 states 'sqrt(2) = sqrt(2)' as if it were a direct substitution or identity, skipping the division step entirely and implying that u·v / ||v|| simplifies directly to sqrt(2) without showing the denominator calculation. More critically, the label on line 3 says 'comp = u·v / ||v||', but the equation shown is just sqrt(2)=sqrt(2), which does not reflect the calculation 10/sqrt(50). This is misleading because it hides the actual arithmetic required to get the answer.
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/vector_projection, checked 2026-10-05 with SymPy 1.14.0.