When to use this calculator
Find the unsigned angle between two nonzero 3D vectors. Zero vectors have no defined direction.
Explore the coordinate & vector calculators for other questions in this topic. For a related calculation, use the Midpoint Calculator: Find the midpoint coordinates between two plane points.
PUTTING IT INTO PRACTICE
Angle Between Vectors: a worked example
The values below are an illustrative example, not a recommendation. Replace them with measurements or assumptions that match your question.
| Input | Example value |
|---|---|
| First vector x | 1 |
| First vector y | 0 |
| First vector z | 0 |
| Second vector x | 0 |
| Second vector y | 1 |
| Second vector z | 0 |
| Result | 90 degrees |
- Check the input labels and enter the example values above.
- Apply the relationship: Angle = acos(dot product/(magnitude one × magnitude two)).
- The result is 90 degrees, rounded for display.
How to interpret your result
Zero vectors have no defined direction.
Keep coordinate units consistent along every axis. Distance is non-negative, while slope and dot product can have signs. A vertical line has undefined slope because the horizontal coordinate change is zero.
Frequently asked questions
Does a zero dot product always mean the vectors are perpendicular?
Two nonzero vectors with zero dot product are perpendicular. The zero vector needs separate treatment because it has no defined direction.
When is the angle between vectors calculator useful?
Find the unsigned angle between two nonzero 3D vectors. Zero vectors have no defined direction.
How do I use the angle between vectors calculator?
Enter first vector x, first vector y, first vector z, second vector x, second vector y, second vector z. The result updates automatically. Use the units shown beside each input; the example values illustrate the method.
Which formula does this calculator use?
Angle = acos(dot product/(magnitude one × magnitude two)).
What should I check before using the result?
Zero vectors have no defined direction.
How are results rounded?
Scalar numbers display up to six decimal places, with scientific notation for very large or small values. Text, fraction, date and schedule outputs use the format described on the tool. Intermediate calculations use JavaScript floating-point arithmetic. Keep sensible precision for your measurements.
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