Calculate the angle between two vectors in 2D or 3D using the dot product formula.
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Dimensions
Vector 1
Vector 2
How it works
The angle between two vectors is found using the dot-product formula: cos(θ) = (v1 · v2) / (|v1| × |v2|). The result is always between 0° and 180°.
Special cases
Parallel vectors give 0°, opposite vectors give 180°, and perpendicular vectors give exactly 90°. A zero-length vector has no defined direction, so the angle is undefined.
Result
The angle between the two vectors is 16.2602° (0.283794 rad).
Step-by-step
Given 2D vectors v1 = (3, 4) and v2 = (4, 3).
Dot product: v1 · v2 = (3 × 4) + (4 × 3) = 24.
|v1| = √(3² + 4²) = 5.
|v2| = √(4² + 3²) = 5.
cos(θ) = 24 / (5 × 5) = 0.96.
θ = arccos(0.96) = 16.2602° (0.283794 rad).
Also in Linear Algebra
Linear Algebra
An angle between vectors calculator uses the dot product formula cos(θ) = (u·v)/(|u||v|) to find the angle between two vectors in 2D or 3D space.
The dot product u·v equals |u||v|cos(θ). Rearranging gives θ = arccos(u·v / (|u||v|)). The result is always between 0° and 180°.
Special cases: perpendicular vectors have a dot product of zero (90°), parallel vectors give 0° or 180°.
θ = arccos((u · v) / (|u| × |v|))
Angle between two vectors.
Frequently asked questions
No — the angle between two vectors is defined as the smallest angle between them, so it is always between 0° and 180° inclusive.
The angle is undefined because the zero vector has no direction. The calculator returns no result in this case.
Related
These related calculators come from the same leaf category, nearby sibling categories, or the same top-level topic.
Calculate the dot product of two vectors in 2D or 3D, with angle and orthogonality detection.
Compute the cross product of two 3D vectors, with magnitude and direction information.
Calculate the magnitude of a 2D or 3D vector with squared magnitude, unit vector, and direction angles.
Normalize a 2D or 3D vector to find its unit vector (magnitude 1) with step-by-step work.
