Dot Product Calculator
Calculate the dot product (inner product) of two vectors.
Inputs
Comma-separated values
Comma-separated values
Dot Product
32.000000
||A||
3.741657
||B||
8.774964
Step by step
Element-wise products summed
Σ(aᵢ × bᵢ)
= 32.000000
||A||
√(Σaᵢ²)
= 3.741657
||B||
√(Σbᵢ²)
= 8.774964
How it works
The dot product of two vectors is the sum of element-wise products. It equals ||A|| × ||B|| × cos(θ) where θ is the angle between them. It is fundamental in neural network computations, attention mechanisms, and similarity measures.
Formula
Dot Product
a · b = sum(a_i * b_i)
- a, b
- Input vectors of equal dimension
Frequently Asked Questions
What does the sign of the dot product tell us?
A positive dot product means the vectors point in roughly the same direction (angle < 90°), zero means they are orthogonal, and negative means they point in opposite directions (angle > 90°).
How is the dot product used in neural networks?
Every neuron computes a dot product between its weight vector and the input vector, followed by a bias addition and activation function. Attention mechanisms also rely heavily on dot products between query and key vectors.