Angle Between Two Vectors Calculator
Angle between two vectors of 2-D or 3-D vectors, with exact fractions and the working.
Change the values and press Calculate to work out your own figures.
Enter the vector components (leave z blank for 2-D).
Show the working
- cos θ = (a · b) ÷ (|a| |b|)
How to use it
Enter the vector components (leave z blank for 2-D).
Enter coordinates as whole numbers, decimals or fractions (3/4). Show the working gives the formula with your numbers.
Key facts
cos θ = (a · b) ÷ (|a| |b|)
Components
Enter the x-, y- and (optionally) z-components; leave z blank for vectors in a plane.
Questions
How is the angle between two vectors worked out?
cos θ = (a · b) ÷ (|a| |b|)
Can I use fractions?
Yes, such as 1/2 or -3/4.
Formulas
cos θ = (a · b) ÷ (|a| |b|)
Sources
Limitations
- Vectors have two or three components (leave z blank for 2-D).
- Earth distances assume the WGS 84 ellipsoid (sea level).