Produto Escalar
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
u·v = Σ uᵢvᵢ
About this calculator
The dot product calculator computes the dot product between two vectors in two or three dimensions. The dot product, also known as the inner product, combines two vectors to yield a scalar. The formula is the sum of the products of corresponding components: u·v = u₁v₁ + u₂v₂ + u₃v₃. For 2D vectors, the third component is zero. This tool is useful for checking angles between vectors, projections, and orthogonality.
You can use this calculator in scenarios such as physics (calculating work done by a force), computer graphics (lighting and reflection), or analytic geometry (determining if vectors are perpendicular). Simply enter the vector coordinates and the result appears instantly. The calculator supports decimal and integer numbers, making it easy for exercises and real problems.
Be careful: ensure both vectors have the same dimension (both 2D or both 3D). The dot product can be negative, indicating an obtuse angle. Remember the result is a scalar, not a vector. For zero vectors, the result is zero. Avoid confusing it with the cross product, which yields a vector.
Frequently asked questions
What does the dot product result mean?
The result is a number indicating the relationship between vectors: positive for acute angle, zero for perpendicular, negative for obtuse angle.
Can I use vectors with more than 3 dimensions?
No, this calculator only supports 2D or 3D. For higher dimensions, a more general tool is needed.
How to calculate the angle between vectors using the dot product?
Use the formula cosθ = (u·v)/(|u||v|). The dot product divided by the product of magnitudes gives the cosine of the angle.
Is the dot product commutative?
Yes, u·v = v·u. The order of vectors does not change the result.
What is the difference between dot product and cross product?
The dot product yields a scalar, while the cross product yields a vector perpendicular to both.