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Angle Between Two 3D Vectors — result sheet
Find the principal angle in degrees between two nonzero three-dimensional Cartesian vectors.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: theta = acos((A dot B)/(|A||B|)) x 180/pi, with the cosine ratio clamped to [-1,1] after finite norm validation.
The principal angle uses the dot product divided by the product of the two nonzero Euclidean norms. The result is reported in degrees from 0 through 180, with a small endpoint clamp only for floating-point drift.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Vector A x — The x component of vector A.; minimum -1000000; maximum 1000000
- Vector A y — The y component of vector A.; minimum -1000000; maximum 1000000
- Vector A z — The z component of vector A.; minimum -1000000; maximum 1000000
- Vector B x — The x component of vector B.; minimum -1000000; maximum 1000000
- Vector B y — The y component of vector B.; minimum -1000000; maximum 1000000
- Vector B z — The z component of vector B.; minimum -1000000; maximum 1000000
Worked example
| Input | Value |
|---|---|
| Vector A x | 1 |
| Vector A y | 0 |
| Vector A z | 0 |
| Vector B x | 0 |
| Vector B y | 1 |
| Vector B z | 0 |
The principal angle between A and B is 90 degrees.
Assumptions and limits
- A and B are nonzero Cartesian 3D vectors expressed in one Euclidean coordinate basis.
- The result is the unsigned principal angle; orientation, rotation direction, and units of the components are not inferred.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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