Angle Between Vectors Calculator

Find the angle between two 2D vectors using the dot product formula.

Last updated: June 2026 | By Patchworkr Team

Vector A

Vector B

Angle

90 deg

1.5708 rad

Calculation

Dot product = 0

|A| = 1

|B| = 1

cos(theta) = dot / (|A| x |B|)

What is the Angle Between Vectors?

The angle is the amount of turning needed to move from one vector to the other, measured as the smaller angle between their directions.

The dot product connects that angle to the lengths of the vectors. When the vectors point in the same direction, the angle is 0 degrees. When they are perpendicular, the angle is 90 degrees. When they point in opposite directions, the angle is 180 degrees.

This is why the calculator uses the dot product formula instead of trying to measure the turn visually. The dot product turns direction into a precise numeric result.

What the Dot Product Tells You

The sign of the dot product reveals how two vectors relate to each other. A positive dot product means the vectors point generally in the same direction, a dot product of zero means they are perpendicular, and a negative dot product means they point in opposite directions.

This is what makes the dot product useful for angle calculations. Once the relationship between the vectors is known, the formula can turn that into a precise angle.

Special Cases

When two vectors point in exactly the same direction, the angle is 0 degrees. When they are perpendicular, the angle is 90 degrees. When they point in opposite directions, the angle is 180 degrees.

These are helpful checkpoints because they make it easier to see whether a calculated answer makes sense.

How to Calculate

The formula compares the dot product to the product of the vector magnitudes. That gives the cosine of the angle, which is then converted back into an actual angle with arccos.

Enter both vectors.
Compute the dot product and magnitudes.
Use cos(theta) = (A dot B) / (|A| x |B|).
Convert the angle to degrees or radians as needed.

Worked Example

A = (3, 4), B = (5, 12) gives theta = 14.25 deg

The result is small because the two vectors point in broadly similar directions. A small angle means the vectors are close to parallel, while a larger angle would mean they point more sharply away from each other.