# Dot Product  One of the most ingenious ideas about the dot product is that there is **no clear geometric interpretation**! Despite it's apparent lack of geometric interpretation, it has *enormous* geometric utility; it has a near monopoly on geometric computation. It turns out that nearly any geometric quantity can be expressed in terms of the inner product. It is that property - to be able to express meaningful geometric quantities - that makes the dot product an essential bridge between geometry and algebra.  ### Wikipedia In mathematics the **dot product** is an [algebraic operation](https://en.wikipedia.org/wiki/Algebraic_operation "Algebraic operation") that takes two equal-length sequences of numbers (usually [coordinate vectors](https://en.wikipedia.org/wiki/Coordinate_vector "Coordinate vector")), and returns a single number. In [Euclidean geometry](https://en.wikipedia.org/wiki/Euclidean_geometry "Euclidean geometry"), the dot product of the [Cartesian coordinates](https://en.wikipedia.org/wiki/Cartesian_coordinates "Cartesian coordinates") of two [vectors](https://en.wikipedia.org/wiki/Vector_(mathematics_and_physics) "Vector (mathematics and physics)") is widely used. It is often called "the" **inner product** of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see [Inner product space](https://en.wikipedia.org/wiki/Inner_product_space "Inner product space") for more). It is often called the **scalar product** as well because it returns a scalar. The dot product may be defined algebraically or geometrically. The geometric definition is based on the notions of angle and distance (magnitude of vectors). The equivalence of these two definitions relies on having a [Cartesian coordinate system](https://en.wikipedia.org/wiki/Cartesian_coordinate_system "Cartesian coordinate system") for Euclidean space.   ### Intuition A nice intuitive way to think about the dot product is that it is a way to measure **similarity**. ### Duality See [my blog post here](https://www.nathanieldake.com/Mathematics/02-Linear_Algebra-02-Linear-Combination-Linear-Transformation-Dot-Product.html#4.-The-Dot-Product), as well as the 3b1b video below. We can summarize the key idea here nicely: > There is a connection between linear transformations that take vectors to numbers, and vectors themselves! ### Algebraic vs Geometric Relationship The key intuition here is that the dot product is a **projection**, and if we take $a \cdot b$ we can view this as projecting $b$ onto a 1-d line defined by $a$.  There is a fantastic article on this [here](https://gregorygundersen.com/blog/2018/06/26/dot-product/#fn:1). ### Dot product vs projection [vectors - What is the difference between the dot product and the scalar projection? - Mathematics Stack Exchange](https://math.stackexchange.com/questions/1064465/what-is-the-difference-between-the-dot-product-and-the-scalar-projection) --- Date: 20211203 Links to: Tags: #todo References: * [Wikipedia](https://en.wikipedia.org/wiki/Dot_product) * [3b1b dot products](https://www.youtube.com/watch?v=LyGKycYT2v0) * [vectors - What is the difference between the dot product and the scalar projection? - Mathematics Stack Exchange](https://math.stackexchange.com/questions/1064465/what-is-the-difference-between-the-dot-product-and-the-scalar-projection) * [Linear Algebra 20g: The Dot Product - One of the Most Brilliant Ideas in All of Linear Algebra - YouTube](https://www.youtube.com/watch?v=QPkKWGq_V0U&t=7s)