Cross product

mathematical operation on two vectors

The cross product is a mathematical operation which can be done between two three-dimensional vectors. It is often represented by the symbol .[1] After performing the cross product, a new vector is formed. The cross product of two vectors is always perpendicular to both of the vectors which were "crossed".[2][3] This means that cross product is normally only valid in three-dimensional space.

Importance of the cross productEdit

Being a vector operation, the cross product is extremely important in all sorts of sciences (particularly physics), engineering, and mathematics. One important example of the cross product involves torque or moment. Another important application involves the magnetic field.

Visualizing the cross product in three dimensionsEdit

Finding the direction of the cross product.

The cross product of   and   is a vector that we shall call  :


The vector   is perpendicular to both   and  . The direction of   is determined by a variation of the right-hand rule. By holding your right hand as shown in the picture, your thumb points in the direction of   (the cross product of   and  ), with the index finger pointing in the direction that   points, and the middle finger pointing in the direction that   points. If the angle between the index and middle fingers is greater than 180°, then you need to turn the hand upside down.

How to calculate the cross product in vector notationEdit

Like any mathematical operation, the cross product can be done in a straightforward way.

Two dimensionsEdit

Since cross products are usually only defined for three-dimensional vectors, the calculation of cross product in two dimensions treat the vectors as if they are vectors on the xy-plane in three dimension.

More specifically, if



where   is just a symbol indicating that the new vector is pointing up (in the z-direction). If one "crosses" two vectors which are both in the xy-plane, then the product, being perpendicular to both vectors, must point in the z direction. If the value of   is positive, then it points out of the page; if its value is negative, then it points into the page.

Three dimensionsEdit

There are two ways to find the cross product of two 3D vectors: with coordinate notation or with angle.

Coordinate notationEdit

Given vectors   and  , where

Then the cross product of   and   is:

With angleEdit

Given vectors   and  , where

Then the cross product of   and   is:
where   is the angle between   and  , ‖a‖ and ‖b‖ are the magnitudes of vectors   and  , and n is a unit vector perpendicular to the plane containing   and  .

Basic properties of the cross productEdit

  • Anti-commutativity:  [2]
  • Distributivity over addition:  [2]
  • Scalar commutavity:  

Related pagesEdit


  1. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-09-06.
  2. 2.0 2.1 2.2 2.3 2.4 Weisstein, Eric W. "Cross Product". Retrieved 2020-09-06.
  3. "Cross Product". Retrieved 2020-09-06.