Dot Product Calculator
Calculate the dot product (scalar product) of two vectors, and see whether they point in a generally similar direction, an opposite direction, or are perpendicular.
About the Dot Product Calculator
The dot product (or scalar product) combines two vectors into a single number by multiplying their corresponding components and adding the results. Unlike the cross product, the dot product works in any number of dimensions and always returns a scalar, not a vector.
Formula: A · B = AxBx + AyBy + AzBz
The sign of the result tells you how the two vectors relate: a positive dot product means the angle between them is less than 90° (acute), a negative dot product means it's greater than 90° (obtuse), and a dot product of exactly zero means the vectors are perpendicular. Leave the z-components at zero to work entirely in 2D.
The dot product shows up throughout physics and engineering — it's used to calculate the work done by a force, to project one vector onto another, and to test whether two directions are at right angles, among many other applications in graphics, mechanics and machine learning.