Vector Angle Calculator
Find the angle between two 2D vectors
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Vector Angle Formula
θ = cos⁻¹((u · v) / (|u||v|))
The angle between vectors comes from the dot product. Dividing the dot product by the product of magnitudes gives the cosine of the angle.
⚠ Both vectors must be nonzero. The result is the smaller angle between the two vectors in degrees.
How Vector Angle Works
The dot product measures how much two vectors point in the same direction. Inverse cosine converts that alignment into an angle.
Dot Product
u·v combines matching components.
Magnitude
Each vector length is required.
Cosine Ratio
dot divided by magnitudes.
Angle
Inverse cosine gives degrees.
💡 Example: For u=(1,0) and v=(0,1), the angle is 90°.
Applications of Vector Angles
Physics Graphics Navigation Geometry
Frequently Asked Questions
What is a vector angle calculator?▼
It calculates the angle between two vectors from their components.
What is the vector angle formula?▼
θ = cos⁻¹((u·v)/(|u||v|)).
How do I use this calculator?▼
Enter the x and y components of both vectors, then click Calculate.
Can a zero vector be used?▼
No. A zero vector has no direction, so the angle is undefined.
Where is vector angle used?▼
It is used in physics, geometry, graphics, navigation, and dot product problems.
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