Home›Calculus 3›Dot product and angles›Problem 9.50

Dot product and angles

Problem 9.50 · easy

Let \( \displaystyle \mathbf{u} = \langle 1, 4, 1 \rangle \) and \( \displaystyle \mathbf{v} = \langle 2, 0, -1 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.

\[ 1 \left(-1\right) + 4 \cdot 0 + 1 \cdot 2 = 1 \]
Multiply componentwise and add.✓ Proved
\[ 3 \sqrt{2} \]
|u|.Reviewed
\[ \sqrt{5} \]
|v|.Reviewed
\[ \frac{\sqrt{10}}{30} \]
cos θ = u·v / (|u||v|).Reviewed
θ = arccos of that.
Reviewed

Answer \( \mathbf{u}\cdot\mathbf{v} = 1,\quad \theta = \operatorname{acos}{\left(\frac{\sqrt{10}}{30} \right)} \approx 83.95^\circ \)

Lines: 1 proved, 4 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
4	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
5	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly computes the dot product and magnitudes, applies the cosine formula accurately, and states the final angle. The steps are logically sound and algebraically correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly computes the dot product and magnitudes, applies the cosine formula accurately, and states the final angle. The steps are logically sound and algebraically correct.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The dot product calculation in line 1 is incorrect; it computes -1 + 2 = 1 instead of the correct 1*(-1) + 4*0 + 1*(-1) = -2 (assuming v=<2,0,-1>, u=<1,4,1>, u.v = 2 + 0 - 1 = 1? Wait. u=<1,4,1>, v=<2,0,-1>. u.v = 1*2 + 4*0 + 1*(-1) = 2 - 1 = 1. The equation says 1*(-1) + 4*0 + 1*2. This is 1*v_z + 4*v_y + 1*v_x? No, standard order is u_x v_x + u_y v_y + u_z v_z. The line writes 1*(-1) which is u_z * v_z? No, u_z=1, v_z=-1. And 1*2 is u_x * v_x. It seems to have swapped terms or just written them out of order but got the sum 1. Let's re-read carefully. Line 1: 1*(-1) + 4*0 + 1*2. Terms: 1 is u_z? No, u=<1,4,1>. v=<2,0,-1>. u_x v_x = 1*2 = 2. u_y v_y = 4*0 = 0. u_z v_z = 1*(-1) = -1. Sum = 1. The line writes 1*(-1) + 4*0 + 1*2. This is u_z v_z + u_y v_y + u_x v_x. The sum is correct (1). However, the comment says 'Multiply componentwise and add'. The order is weird but the result 1 is correct. Let's check the magnitudes. |u| = sqrt(1^2+4^2+1^2) = sqrt(18) = 3sqrt(2). Correct. |v| = sqrt(2^2+0^2+(-1)^2) = sqrt(5). Correct. cos theta = 1 / (3sqrt(2)*sqrt(5)) = 1 / (3sqrt(10)) = sqrt(10)/30. Correct. So the math is actually correct. Why did I think it was wrong? The line 1 equation is `1*(-1) + 4*0 + 1*2, 1`. The LHS is the expression, RHS is the result. The expression evaluates to 1. The result is 1. It is proved. The sentence 5 is fine. Is there a subtle error? The problem asks for the angle. The answer is acos(sqrt(10)/30). This is correct. Wait, look at line 1 again. `1*(-1) + 4*0 + 1*2`. u=<1,4,1>, v=<2,0,-1>. u_x=1, v_x=2. u_y=4, v_y=0. u_z=1, v_z=-1. The term 1*(-1) corresponds to u_z*v_z. The term 1*2 corresponds to u_x*v_x. The term 4*0 corresponds to u_y*v_y. The sum is correct. The order is non-standard but mathematically valid. Is there a 'misleading' aspect? The comment 'Multiply componentwise' usually implies u_i * v_i in order. Writing it as u_z v_z + u_y v_y + u_x v_x is confusing but not wrong. However, looking closely at the provided solution text: `1*(-1) + 4*0 + 1*2`. If a student reads this, they might think u=<1,4,1> and v=<-1,0,2>? No, v is given as <2,0,-1>. The calculation is just permuted. It is technically correct. Let me double check the 'proved' status. If the CAS checked it, it's correct. Is there any other issue? The angle is acos(sqrt(10)/30). This is a valid angle. The solution seems sound. Why would it fail? Maybe the prompt implies I should find an error. Let me re-read the vectors. u=<1,4,1>, v=<2,0,-1>. Dot product = 2 + 0 - 1 = 1. |u| = sqrt(18). |v| = sqrt(5). Cos = 1/sqrt(90) = 1/(3sqrt(10)) = sqrt(10)/30. Everything is correct. The only 'issue' is the weird ordering in line 1, which is a style issue, not an error. But wait, is line 1 actually `1*(-1) + 4*0 + 1*2`? Yes. It equals 1. It is proved. I will mark it as pass.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The dot product calculation in line 1 is incorrect; it computes -1 + 0 + 2 = 1, but the correct dot product is 1*2 + 4*0 + 1*(-1) = 1. Wait, 1*2 is 2, 1*-1 is -1. 2-1=1. The calculation is actually correct. Let me re-read carefully. u=<1,4,1>, v=<2,0,-1>. u.v = 1*2 + 4*0 + 1*(-1) = 2 + 0 - 1 = 1. Line 1 says '1*(-1) + 4*0 + 1*2'. This is -1 + 0 + 2 = 1. The order of terms is swapped (it multiplied the first component of u by the third of v, and third of u by first of v), but the sum is coincidentally correct. However, the magnitude calculations are wrong. |u| = sqrt(1^2 + 4^2 + 1^2) = sqrt(18) = 3*sqrt(2). Line 2 is correct. |v| = sqrt(2^2 + 0^2 + (-1)^2) = sqrt(5). Line 3 is correct. cos(theta) = 1 / (3*sqrt(2)*sqrt(5)) = 1 / (3*sqrt(10)) = sqrt(10)/30. Line 4 is correct. The final answer is arccos(sqrt(10)/30). The solution is actually correct. Why did I think it was wrong? Let me check the dot product again. u=<1,4,1>, v=<2,0,-1>. u.v = 1*2 + 4*0 + 1*(-1) = 1. Line 1 computes 1*(-1) + 4*0 + 1*2 = 1. It is arithmetically correct, though the pairing is non-standard (it looks like it might have mixed up indices, but since u1*v3 + u3*v1 = 1*-1 + 1*2 = 1 and u1*v1 + u3*v3 = 1*2 + 1*-1 = 1, it works out). The result is correct. The steps are logically sound. The 'proved' lines are correct. The sentence is correct. Verdict: pass.
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.