Math
Vector dot product
Compute the scalar dot product of two three-dimensional vectors.
Calculators / Math / Angle between vectors
Angle between vectors calculator
Find the smaller angle between two nonzero three-dimensional vectors.
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Angle12.93degrees
Calculator controls
How to calculate
Find the smaller angle between two nonzero three-dimensional vectors.
Angle = acos((((ax*bx+ay*by+az*bz)/sqrt((ax^2+ay^2+az^2)*(bx^2+by^2+bz^2))-1+abs((ax*bx+ay*by+az*bz)/sqrt((ax^2+ay^2+az^2)*(bx^2+by^2+bz^2))+1))/2+1-abs(((ax*bx+ay*by+az*bz)/sqrt((ax^2+ay^2+az^2)*(bx^2+by^2+bz^2))-1+abs((ax*bx+ay*by+az*bz)/sqrt((ax^2+ay^2+az^2)*(bx^2+by^2+bz^2))+1))/2-1))/2)*180/pi degrees
Worked example
Vector A x: 1; Vector A y: 2; Vector A z: 3; Vector B x: 4; Vector B y: 5; Vector B z: 6.
Angle: 12.93 degrees.
Assumptions and limits
The angle is between zero and 180 degrees. Both vectors must be nonzero; the dot-product cosine is limited at both rounding boundaries.
Results are rounded for display; calculations use unrounded values. Read our calculation methodology.
Understanding the result
Find the smaller angle between two nonzero three-dimensional vectors.
When this tool is useful
Angle = acos((((ax*bx+ay*by+az*bz)/sqrt((ax^2+ay^2+az^2)*(bx^2+by^2+bz^2))-1+abs((ax*bx+ay*by+az*bz)/sqrt((ax^2+ay^2+az^2)*(bx^2+by^2+bz^2))+1))/2+1-abs(((ax*bx+ay*by+az*bz)/sqrt((ax^2+ay^2+az^2)*(bx^2+by^2+bz^2))-1+abs((ax*bx+ay*by+az*bz)/sqrt((ax^2+ay^2+az^2)*(bx^2+by^2+bz^2))+1))/2-1))/2)*180/pi degrees
Understanding your inputs
| Input | What to enter |
|---|---|
| Vector A x | Enter a number of at least -1000000000000 and no more than 1000000000000. |
| Vector A y | Enter a number of at least -1000000000000 and no more than 1000000000000. |
| Vector A z | Enter a number of at least -1000000000000 and no more than 1000000000000. |
| Vector B x | Enter a number of at least -1000000000000 and no more than 1000000000000. |
| Vector B y | Enter a number of at least -1000000000000 and no more than 1000000000000. |
| Vector B z | Enter a number of at least -1000000000000 and no more than 1000000000000. |
Frequently asked questions
What assumptions does this calculation use?
The angle is between zero and 180 degrees. Both vectors must be nonzero; the dot-product cosine is limited at both rounding boundaries.
How is the result calculated?
Angle = acos((((ax*bx+ay*by+az*bz)/sqrt((ax^2+ay^2+az^2)*(bx^2+by^2+bz^2))-1+abs((ax*bx+ay*by+az*bz)/sqrt((ax^2+ay^2+az^2)*(bx^2+by^2+bz^2))+1))/2+1-abs(((ax*bx+ay*by+az*bz)/sqrt((ax^2+ay^2+az^2)*(bx^2+by^2+bz^2))-1+abs((ax*bx+ay*by+az*bz)/sqrt((ax^2+ay^2+az^2)*(bx^2+by^2+bz^2))+1))/2-1))/2)*180/pi degrees