Dot product and angles
Problem 9.48 · easy
Let \( \displaystyle \mathbf{u} = \langle 5, -1, 2 \rangle \) and \( \displaystyle \mathbf{v} = \langle 4, 2, -4 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
- \[ 2 \left(-4\right) - 1 \cdot 2 + 5 \cdot 4 = 10 \]Multiply componentwise and add.✓ Proved
- \[ \sqrt{30} \]|u|.✓ Proved
- \[ 6 \]|v|.✓ Proved
- \[ \frac{\sqrt{30}}{18} \]cos θ = u·v / (|u||v|).✓ Proved
- θ = arccos of that.
Answer \( \mathbf{u}\cdot\mathbf{v} = 10,\quad \theta = \operatorname{acos}{\left(\frac{\sqrt{30}}{18} \right)} \approx 72.28^\circ \)
Lines: 4 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 |
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 2026-09-26 with SymPy 1.14.0.