Angle Between Vectors
Find the angle between two 2D or 3D vectors using the dot product formula. Free online vector angle calculator with cosine similarity and magnitude results.
About This Calculator
This calculator computes the angle between two 2D or 3D vectors using the dot product formula: cos(theta) = (v·w) / (|v| x |w|). Enter the components of both vectors and get the angle in degrees, the cosine of the angle, and the dot product value.
Vector angle calculations are fundamental in physics (force analysis, projectile motion), computer graphics (lighting, camera angles), machine learning (cosine similarity for text analysis), and engineering (structural analysis, robotics).
Frequently Asked Questions
How do you find the angle between two vectors?
Use the dot product formula: cos(theta) = (v·w) / (|v| x |w|). The angle theta = arccos((v·w) / (|v| x |w|)).
What is the dot product?
The dot product of v = (x1, y1, z1) and w = (x2, y2, z2) is v·w = x1x2 + y1y2 + z1z2. It is a scalar representing the product of magnitudes and cosine of the angle between them.
What does a 90 deg angle mean?
A 90 deg angle means the vectors are orthogonal (perpendicular). The dot product of orthogonal vectors is always zero.
What does a 0 deg angle mean?
A 0 deg angle means the vectors are parallel and point in the same direction. cos(theta) = 1 and the dot product equals the product of magnitudes.
Can I calculate angle between 2D vectors?
Yes. Enter x and y components and leave z empty (defaults to 0). The calculator handles both 2D and 3D vectors.
Is this tool free?
Yes, all calculators on Calculy are completely free to use with no registration required.