VEC011 · Vectors · Classes 11–12
Angle between vectors
The angle between two vectors comes from the dot product.
cos θAngle between vectors
📐 Formula
cos θ = (u · v) / (|u||v|)
✏️ Worked example
(3, 1) and (1, 2): cos θ = 5/(√10 × √5) = 1/√2, so θ = 45°.
🌍 Where it is used
Angles between forces, roads or molecules.
❓ Quick check
Angle between (1, 0) and (0, 1)?
🏋️ Practice
EasyAngle between (1, 1) and (1, 0)?
Answer: 45°
MediumAngle between v and −v?
Answer: 180°
ChallengeIf u · v > 0, the angle is…?
Answer: acute