====== Cross Product ======


<tabbox Intuitive> 

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
</note>
  
<tabbox Concrete> 
{{ :basic_tools:vector_calculus:crossproduct.png?nolink|}}

<blockquote>

  * Dot product, the interactions between similar dimensions (x*x y*y, z*z)
  * Cross product, the interactions between different dimensions (x*y,y*z, z*x, etc.)

The dot product (vec(a) · vec(b)) measures similarity because it only accumulates interactions in matching dimensions. It’s a simple calculation with 3 components.

The cross product (written vec(a) times vec(b)) has to measure a half-dozen “cross interactions”. The calculation looks complex but the concept is simple: accumulate 6 individual differences for the total.

<cite>[[https://betterexplained.com/articles/vector-calculus-understanding-the-dot-product/|Vector Calculus: Understanding the Dot Product]] by Kalid Azad</cite></blockquote>

**Recommended Resources:**

  * [[https://betterexplained.com/articles/vector-calculus-understanding-the-dot-product/|Vector Calculus: Understanding the Dot Product]] by Kalid Azad
  * A Student's Guide to Vectors and Tensors by Daniel A. Fleisch
 
<tabbox Abstract> 

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

  
<tabbox Why is it interesting?> 


The dot product is a tool that we can use to combine two vectors and get another vector out. This number describes the plane spanned by the two vectors. The magnitude of the resulting vector is the area of the plane, and the direction denotes the orientation of the plane, because it is perpendicular to it. 

This is an extremely useful concept and used in almost any physical theory, like for example, [[models:classical_electrodynamics|electrodynamics]]. Moreover, many other important tools, like the [[basic_tools:vector_calculus:curl]] are defined with the help of the cross product. 


</tabbox>