Parametric, Polar & Vectors · Vector Operations
Angle Between Vectors
The angle between two vectors is found using the dot product divided by the product of their magnitudes.
Variables
| Symbol | Name | Unit |
|---|---|---|
| u1 | u x-component | — |
| u2 | u y-component | — |
| v1 | v x-component | — |
| v2 | v y-component | — |
Worked examples
Find the angle between ⟨1, 0⟩ and ⟨1, 1⟩.
- u · v = 1. |u| = 1, |v| = √2.
- cos θ = 1/√2 → θ = π/4 = 45°
Answer: π/4 (45°)
Find the angle between ⟨1, 2, 3⟩ and ⟨-1, 0, 1⟩.
- u · v = -1+0+3 = 2. |u| = √14, |v| = √2.
- cos θ = 2/√28 = 2/(2√7) = 1/√7
- θ = arccos(1/√7) ≈ 67.79°
Answer: arccos(1/√7) ≈ 67.8°
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