Angle Between Vectors
Find the angle from the dot product · 벡터 사이 각
1. What Is the Angle Between Two Vectors?
When two vectors start from the same point, they open up a single angle θ between them, measured from 0° (pointing the same way) to 180° (pointing opposite ways). Finding that angle is one of the headline jobs of the dot product.
The angle tells you how aligned two directions are — central to physics (work, forces), graphics (lighting), and similarity measures.
두 벡터를 같은 점에서 그리면 0°(같은 방향)부터 180°(반대 방향)까지의 각 θ가 생깁니다. 이 각을 구하는 것은 내적의 핵심 용도이며, 두 방향이 얼마나 정렬되어 있는지를 나타냅니다.
2. The Formula & a Worked Example
Rearrange the geometric form of the dot product, a · b = |a| |b| cos θ, to isolate the angle:
cos θ = (a · b) / (|a| |b|), so θ = arccos((a · b) / (|a| |b|))
Worked example. For a = (1, 1) and b = (1, 0):
- Dot product: a · b = (1)(1) + (1)(0) = 1
- Magnitudes: |a| = √2, |b| = 1
- cos θ = 1 / (√2 · 1) = 1/√2 ≈ 0.7071, so θ = arccos(1/√2) = 45°
a · b = |a| |b| cos θ 를 변형하면 cos θ = (a · b)/(|a| |b|), θ = arccos(...) 입니다. 예: a = (1, 1), b = (1, 0) → a · b = 1, |a| = √2, |b| = 1 → cos θ = 1/√2 → θ = 45°.
3. Reading the Sign
Because |a| and |b| are always positive, the sign of a · b alone classifies the angle without computing arccos:
- a · b > 0 → 0° ≤ θ < 90°, the angle is acute.
- a · b = 0 → θ = 90°, the vectors are perpendicular.
- a · b < 0 → 90° < θ ≤ 180°, the angle is obtuse.
|a|, |b|는 항상 양수이므로 a · b의 부호만으로 각을 분류할 수 있습니다: 양수면 예각(θ < 90°), 0이면 직교(θ = 90°), 음수면 둔각(θ > 90°).
4. Special Cases & Key Properties
- Perpendicular. θ = 90° exactly when a · b = 0 (the orthogonality test).
- Same direction. θ = 0° when b is a positive scalar multiple of a.
- Opposite direction. θ = 180° when b is a negative scalar multiple of a.
- Range. The angle is always in 0° ≤ θ ≤ 180°; order does not matter since a · b = b · a.
a · b = 0 이면 θ = 90°(직교), b가 a의 양의 스칼라배면 θ = 0°, 음의 스칼라배면 θ = 180°. 각의 범위는 0°~180°이며, a · b = b · a 이므로 순서는 무관합니다.
5. Frequently Asked Questions
How do you find the angle between two vectors? Compute θ = arccos((a · b) / (|a| |b|)) — divide the dot product by the product of the magnitudes, then take the inverse cosine.
How do you know if two vectors are perpendicular? Their dot product is 0, which forces θ = 90°.
Can the angle be more than 180°? No. The angle between two vectors is always reported in the range 0° to 180°.
θ = arccos((a · b)/(|a| |b|))로 구합니다. 내적이 0이면 두 벡터는 수직(θ = 90°)입니다. 두 벡터 사이 각은 항상 0°~180° 범위로 나타냅니다.
Ready to practice? Drill angle problems and the rest of linear algebra on C:Vector, or review the full vector reference and the related dot product, magnitude, and projection.