Dot product

What is it?

Formulas

$$u \cdot v = \sum_{i=1}^n u_i v_i = \norm u\,\norm v \cos\theta$$

$$\operatorname{proj}_u v = \frac{u \cdot v}{u \cdot u}\, u$$

projection

Where it shows up in computing

• Lighting and shading★★★★★fundamentalComputer graphics

  Diffuse lighting is $\max(0, n \cdot l)$, the dot product of the unit normal and the light direction.

• Recommender systems★★★★★frequentOptimization and systems

  Predicted affinity between a user and an item is the dot product of their embedding vectors.

Where it shows up in AI

• Neural networks★★★★★fundamentalAI and machine learning

  Each neuron computes $w \cdot x + b$ before its activation.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

What depends on it

• Matrices and linear maps
• Directional derivative