Angle Between Vectors Calculator
The vector-angle calculator finds the smaller angle between two non-zero three-dimensional vectors using their dot product and magnitudes.
How to use this calculator
- Enter three components for each vector. The tool returns an angle between 0° and 180°.
- Check the units and definitions before comparing results with another source.
- Change one input at a time when comparing scenarios, and keep the reporting period and assumptions consistent.
Formula used by the vector angle calculator
θ = arccos((a · b)/(|a||b|))
- θ = arccos((a · b)/(|a||b|))
- A zero vector has no direction, so its angle with another vector is undefined. Floating-point rounding may affect the last displayed digit.
- Inputs are checked for required values and valid ranges; displayed results are rounded for readability.
Worked examples
Worked example
A = (1, 0, 0) and B = (0, 1, 0).
Their dot product is zero, so the angle is arccos(0) = 90°.
Their dot product is zero, so the angle is arccos(0) = 90°.
Understanding the result
The vector-angle calculator finds the smaller angle between two non-zero three-dimensional vectors using their dot product and magnitudes.
A zero vector has no direction, so its angle with another vector is undefined. Floating-point rounding may affect the last displayed digit.
Important notes
A zero vector has no direction, so its angle with another vector is undefined. Floating-point rounding may affect the last displayed digit.
The result applies to the numbers and assumptions entered here; verify specifications for safety-critical or regulated uses.
Frequently asked questions
What is a 90-degree vector angle?
The vectors are perpendicular when their dot product is zero and both vectors are non-zero.
Can the angle exceed 180 degrees?
This calculator reports the smaller angle between vectors, which is from 0° to 180°.
How do I use this calculator?
Enter three components for each vector. The tool returns an angle between 0° and 180°.
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This page provides a calculation tool plus explanatory information so you can understand what the result means and when the underlying method may not fit your situation.