Dot Product Calculator
The dot product is a fundamental operation in vector mathematics that combines two vectors to produce a scalar value. It measures how much two vectors point in the same direction and is essential in physics, engineering, and graphics. This calculator computes the dot product for 2D and 3D vectors and also calculates the angle between them, helping you understand the geometric relationship between your vectors.
Dot product formula
A · B = a1*b1 + a2*b2 + a3*b3
cos(θ) = (A · B) / (|A| * |B|)
θ = arccos((A · B) / (|A| * |B|))
Dot product calculator: frequently asked questions
What is the dot product?
The dot product (or scalar product) of two vectors A and B is a scalar value calculated as A · B = a1*b1 + a2*b2 + a3*b3 (and so on for more dimensions). It combines the magnitudes and the angle between the vectors. The result is positive if the vectors point in similar directions, negative if opposite, and zero if perpendicular.
How do you find the angle between two vectors?
The angle θ between vectors A and B can be found using cos(θ) = (A · B) / (|A| * |B|). Therefore θ = arccos((A · B) / (|A| * |B|)). This gives the angle in radians; multiply by 180/π to get degrees.
When is the dot product zero?
The dot product is zero when the two vectors are perpendicular (at right angles, 90 degrees). This is a useful property for checking orthogonality without computing angles.
What does a negative dot product mean?
A negative dot product means the angle between the vectors is greater than 90 degrees. The vectors point more in opposite directions than in the same direction. The more negative the dot product, the more opposite the directions.
What are applications of the dot product?
The dot product is used in physics to compute work (force dot displacement), in computer graphics for lighting calculations, in machine learning for similarity measures, and in engineering to analyze forces and projections.
Official sources
- Khan Academy: Dot product.
- Wolfram MathWorld: Dot product.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.