Vector Angle Calculator
Calculate the angle between two vectors using the dot product formula.
About Vector Angle Calculator
The angle θ between two vectors is found from the dot product formula: cos(θ) = (A·B) / (||A|| · ||B||). The result is always between 0° and 180°. Orthogonal vectors yield 90°, parallel vectors yield 0° or 180°.
Formula
Angle Formula
θ = arccos(A·B / (||A|| · ||B||))
- A
- First vector
- B
- Second vector
- θ
- Angle between them
Worked example
With the default values below, Vector Angle Calculator returns Angle (degrees) of 12.9332. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Vector A (comma or space separated)
- 1, 2, 3
- Vector B (comma or space separated)
- 4, 5, 6
Results
- Angle (degrees)
- 12.9332
- Angle (radians)
- 0.225726
- cos(θ)
- 0.974632
Step by step
- 1A= [1, 2, 3]
- 2B= [4, 5, 6]
- 3A · B= 32
- 4||A||= 3.741657
- 5||B||= 8.774964
- 6cos(θ) = A·B / (||A||·||B||)= 0.974632
- 7θ= 12.9332° (0.225726 rad)
Assumptions and limits
- The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
Can the angle be greater than 180°?
No. The angle from the dot product formula is always in [0°, 180°]. For oriented angles in 2D you would need atan2.
What angle do orthogonal vectors make?
Exactly 90°. The cosine of 90° is zero, which means the dot product is zero.