Free · Mathematics models
Angle Between Two Nonzero Vectors
Find the principal angle for vectors entered as components or starting and ending coordinates in two or three dimensions.
- Formula & worked example
- Private in your browser
- No signup
Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
The directions (1,0,0) and (0,1,0) form a 90-degree angle.
cos(angle)=dot product/(product of magnitudes); point representations first subtract their endpoints.
Worked example
Enter these known values and leave the other values blank.
- x
- 1
- y
- 0
- z
- 0
- x
- 0
- y
- 1
- z
- 0
- α
- 90 deg
Assumptions and limitations
- Each vector is nonzero. All eight component/point and dimensional representation combinations are preserved. Components are coordinates in one common basis.
- The principal angle lies from zero to 180 degrees. Scaling either vector by a positive factor does not change it; reversing one vector changes it to its supplement.
- This model does not assign a signed rotation angle or apply a coordinate-system transformation.
- Clear the unknown numerical field before using a supported inverse; selection controls retain their captured branches.
Common questions
What assumptions matter for this model?
Each vector is nonzero. All eight component/point and dimensional representation combinations are preserved. Components are coordinates in one common basis.
What does this model leave out?
This model does not assign a signed rotation angle or apply a coordinate-system transformation.
References
Model requirements: The first component vector must be nonzero. The second component vector must be nonzero. Two-dimensional component vectors must also be nonzero. Authored accessible explanations preserve each captured diagram mode and its geometric relationships.