# The Inner Product as a Decision Rule

DevFeed: [The Inner Product as a Decision Rule](<https://devfeed.tech/articles/the-inner-product-as-a-decision-rule-40410.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2017/05/22/the-inner-product-as-a-decision-rule/>)

Published: 2017-05-22T08:00:00Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [math](<https://devfeed.tech/topics/math.md>), [Code](<https://devfeed.tech/topics/code.md>)

Tags: [arithmetic](<https://devfeed.tech/tags/arithmetic.md>), [inner-product](<https://devfeed.tech/tags/inner-product.md>), [javascript](<https://devfeed.tech/tags/javascript.md>), [linear-algebra](<https://devfeed.tech/tags/linear-algebra.md>), [machine-learning](<https://devfeed.tech/tags/machine-learning.md>), [math](<https://devfeed.tech/tags/math.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [programming](<https://devfeed.tech/tags/programming.md>), [projection](<https://devfeed.tech/tags/projection.md>), [support-vector-machines](<https://devfeed.tech/tags/support-vector-machines.md>), [vectors](<https://devfeed.tech/tags/vectors.md>)

## AI overview

This article explains how the standard inner product, or dot product, acts as a geometric decision rule. It shows how the sign of the product determines which side of a line through the origin a vector lies on, while zero indicates that the vector lies on the line. The explanation connects this behavior to vector projection.

## Source excerpt

The standard inner product of two vectors has some nice geometric properties. Given two vectors $ x, y \in \mathbb{R}^n$, where by $ x_i$ I mean the $ i$-th coordinate of $ x$, the standard inner product (which I will interchangeably call the dot product) is defined by the formula $$\displaystyle \langle x, y \rangle = x_1 y_1 + \dots + x_n y_n$$ This formula, simple as it is, produces a lot of interesting geometry.