Dot Product Calculator
Calculate the dot product of two 2D or 3D vectors and the angle between them.
Vector A
Vector B
a · b =
|A| =
|B| =
Angle θ =
The vectors are orthogonal (perpendicular), because their dot product is 0.
The Dot Product Formula
The dot product (also called the scalar product) of two vectors is the sum of the products of their corresponding components. For vectors A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃):
- Dot product: a · b = a₁·b₁ + a₂·b₂ + a₃·b₃
- Magnitude of A: |A| = √(a₁² + a₂² + a₃²)
- Magnitude of B: |B| = √(b₁² + b₂² + b₃²)
- Angle: θ = arccos( (a · b) / (|A| · |B|) )
In two dimensions the third component is simply dropped, so a · b = a₁·b₁ + a₂·b₂.
Geometric Meaning
Geometrically, the dot product equals |A| · |B| · cos(θ), where θ is the angle between the two vectors. It measures how much one vector extends in the direction of another, which is why it is closely tied to the idea of projection: the scalar projection of A onto B is (a · b) / |B|.
Tip: A positive dot product means the vectors point in broadly the same direction (angle < 90°), while a negative dot product means they point in broadly opposite directions (angle > 90°).
Orthogonality
Two non-zero vectors are orthogonal (perpendicular) precisely when their dot product is zero. This is because cos(90°) = 0, so a · b = |A| · |B| · cos(90°) = 0. Orthogonality is a foundational concept in geometry, physics, and linear algebra, used for everything from defining coordinate axes to building orthonormal bases.
Related Calculators
Note: This calculator provides mathematical calculations for vector dot products, magnitudes, and angles based on the formulas described. Results are accurate to two decimal places. While we strive for accuracy, please verify important calculations independently. This tool is for educational and informational purposes and should not be the sole basis for academic or professional decisions.