Home›Calculus 3›Dot product and angles›Problem 9.13

Dot product and angles

Problem 9.13 · easy

Let \( \displaystyle \mathbf{u} = \langle 4, -4, 4 \rangle \) and \( \displaystyle \mathbf{v} = \langle 2, -5, -5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.

\[ 4 \left(-5\right) + 4 \cdot 2 - 4 \left(-5\right) = 8 \]
Multiply componentwise and add.✓ Proved
\[ 4 \sqrt{3} \]
|u|.
\[ 3 \sqrt{6} \]
|v|.
\[ \frac{\sqrt{2}}{9} \]
cos θ = u·v / (|u||v|).
θ = arccos of that.

Answer \( \mathbf{u}\cdot\mathbf{v} = 8,\quad \theta = \operatorname{acos}{\left(\frac{\sqrt{2}}{9} \right)} \approx 80.96^\circ \)

Lines: 1 proved, 4 not checked. 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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
3	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
4	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
5	Not checked	—	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the law of cosines on the triangle u, v, u − v gives the same angle

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The dot product calculation in step 1 is incorrect; it computes 8 instead of the correct value -36. Consequently, the angle is wrong.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The dot product calculation in step 1 is incorrect; it computes 8 instead of the correct value -36. Consequently, the angle is wrong.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The dot product calculation in step 1 is incorrect. The correct value is 4(2) + (-4)(-5) + 4(-5) = 8 + 20 - 20 = 8, but the expression shown (4*(-5) + 4*2 - 4*(-5)) evaluates to 8 only by coincidence of signs if interpreted loosely, but strictly 4*(-5) is -20, 4*2 is 8, -4*(-5) is 20, sum is 8. Wait, let's re-read carefully. u=<4,-4,4>, v=<2,-5,-5>. u.v = 4*2 + (-4)*(-5) + 4*(-5) = 8 + 20 - 20 = 8. The equation line says: 4*(-5) + 4*2 - 4*(-5). This corresponds to components: u_z*v_z? No. u_x*v_z? No. It looks like a jumbled mess of terms. 4*(-5) is u_x*v_y? No, v_y is -5. u_x is 4. So 4*(-5) is u_x*v_y. 4*2 is u_x*v_x? No, u_x is 4, v_x is 2. -4*(-5) is u_y*v_y? u_y is -4, v_y is -5. So the terms are u_x*v_y + u_x*v_x + u_y*v_y. It is missing the z-component u_z*v_z = 4*(-5) = -20. The sum of the written terms is -20 + 8 + 20 = 8. The correct sum is 8 + 20 - 20 = 8. The numerical result 8 is correct, but the expression written is not the dot product formula. It's a 'style' or 'misleading' error if the result is right, but let's look closer. Actually, 4*(-5) + 4*2 - 4*(-5) = -20 + 8 + 20 = 8. The correct dot product is 8. So the number is right. However, the expression is nonsensical as a representation of u·v. Is it an error? It's a bad explanation. But wait, look at step 2 and 3. |u| = sqrt(16+16+16) = sqrt(48) = 4*sqrt(3). Correct. |v| = sqrt(4+25+25) = sqrt(54) = 3*sqrt(6). Correct. cos theta = 8 / (4sqrt(3)*3sqrt(6)) = 8 / (12*sqrt(18)) = 8 / (12*3*sqrt(2)) = 8 / (36*sqrt(2)) = 2 / (9*sqrt(2)) = sqrt(2)/9. Correct. The final answer is acos(sqrt(2)/9). The only issue is step 1's expression. It calculates 8, but via a weird combination of terms that doesn't match the standard definition. It's misleading/wrong setup. But is it an 'error' in the solution? The result is correct. The prompt asks to find what is wrong. The expression in step 1 is not the dot product. It's a calculation error in the *representation*, even if the arithmetic sum happens to be correct. A student would learn that you can mix up components and still get the right answer? That's misleading. Let's call it misleading.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The dot product calculation in step 1 is incorrect; it yields 8, but the correct value is 4(2) + (-4)(-5) + 4(-5) = 8 + 20 - 20 = 8. Wait, let me re-calculate. u = <4, -4, 4>, v = <2, -5, -5>. u.v = 4*2 + (-4)*(-5) + 4*(-5) = 8 + 20 - 20 = 8. The calculation is actually correct. Let me re-read the vector components carefully. u = <4, -4, 4>, v = <2, -5, -5>. Dot product: 4*2 + (-4)*(-5) + 4*(-5) = 8 + 20 - 20 = 8. Magnitude u: sqrt(16+16+16) = sqrt(48) = 4*sqrt(3). Magnitude v: sqrt(4+25+25) = sqrt(54) = 3*sqrt(6). cos(theta) = 8 / (4*sqrt(3) * 3*sqrt(6)) = 8 / (12 * sqrt(18)) = 8 / (12 * 3*sqrt(2)) = 8 / (36*sqrt(2)) = 2 / (9*sqrt(2)) = sqrt(2)/9. The math is correct. Why did I think it was wrong? Let me check the stated answer. acos(sqrt(2)/9). The solution derives sqrt(2)/9. The steps are correct. The sentence in step 5 is vague but not false. Is there a hidden error? Let's check the dot product again. 4*2=8. -4*-5=20. 4*-5=-20. Sum=8. Correct. |u| = sqrt(48) = 4sqrt(3). Correct. |v| = sqrt(54) = 3sqrt(6). Correct. Product of magnitudes = 12 sqrt(18) = 12 * 3 sqrt(2) = 36 sqrt(2). Ratio = 8 / 36 sqrt(2) = 2 / 9 sqrt(2) = sqrt(2)/9. Correct. The solution is sound.
gpt-oss:20b: pass 2026-09-26

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/dot_product_angle, checked 2026-10-13 with SymPy 1.14.0.