Two Dimensional Vector Angle Calculation
Using this Online Calculator, we can Calculate the Angle for two vectors v and w
Calculator
Interactive Vector Diagram
2D Vector Angle Formula with Solved Example
Angle Between Two Vectors Formula
cos θ = (A · B) / (|A| × |B|)
Dot Product
A · B = (x₁ × x₂) + (y₁ × y₂)
Magnitude
|A| = √(x₁² + y₁²)
|B| = √(x₂² + y₂²)
Final Angle
θ = cos⁻¹[(A · B) / (|A| × |B|)]
Solved Example
Vector A = (3, 4)
Vector B = (5, 2)
1
Find Dot Product
A · B = (3 × 5) + (4 × 2)
A · B = 23
2
Find Magnitude of Vector A
|A| = √(3² + 4²)
|A| = 5
3
Find Magnitude of Vector B
|B| = √(5² + 2²)
|B| = 5.385
4
Substitute Values
cos θ = 23 / (5 × 5.385)
cos θ = 0.854
5
Find Angle
θ = cos⁻¹(0.854)
θ ≈ 31.6°