Spatial angle tools compare directions, planes, or three-dimensional turns through dot products and geometric orientation.
Use it to measure orientation in the stated dimension and distinguish an ordinary angle from a directed one.
The relationship
Write the rule before calculating
cos θ = (a·b)/(‖a‖‖b‖), with 0° ≤ θ ≤ 180°.
See the structure
What the calculation is doing
u ↗θv →cos θ = (a·b)/(‖a‖‖b‖), with 0° ≤ θ ≤ 180°.
Worked interpretation
Read the result in context
Vectors (1,0) and (0,1) meet at 90°.
Interpret with care
Important boundary
A zero vector or degenerate plane has no usable direction for an angle calculation.
Use the Mathematics collection to move between connected concepts without duplicating the calculation.
Quick guide
How to use this calculator
Choose the calculation mode when the page supports more than one direction.
Enter the known measurements using the labels and angle units shown.
Read the primary result first, then use the supporting rows to verify the convention or relationship.
Calculation method
Apply the selected angle relationship
cos θ = (a·b)/(‖a‖‖b‖), with 0° ≤ θ ≤ 180°.
The calculator validates geometric domains and reports degenerate vectors, impossible measurements, or unsupported ranges explicitly.
Worked example
Worked example
Vectors (1,0) and (0,1) meet at 90°.
cos θ = (a·b)/(‖a‖‖b‖), with 0° ≤ θ ≤ 180°.
Supported inputs
Precision and limits
Angle conventions
Positive Cartesian rotations are counterclockwise. North-referenced azimuths run clockwise. Unoriented line and plane calculators return the smaller acute or right angle; directed vector and dihedral tools can return up to 180°.
Numerical range
Inputs may be zero or have absolute magnitude from 1e-100 through 1e100. General angle magnitudes are limited to 1,000,000,000 degrees so periodic reduction remains meaningful in double-precision arithmetic.
Precision
Calculations use finite double-precision arithmetic and normally display 12 significant digits. Results are numerical unless the page explicitly displays a fixed geometric identity.
