# Cross product

The cross product is a mathematical operation which can be done between two vectors. 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". This means that the cross product must always be used in 3-Dimensional space.

## Importance of the cross product

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 dimensions

Finding the direction of the cross product.

The cross product of ${\displaystyle {\vec {a}}}$ and ${\displaystyle {\vec {b}}}$ is a vector that we shall call ${\displaystyle {\vec {c}}}$:

${\displaystyle {\vec {c}}={\vec {a}}\times {\vec {b}}}$

Then, the magnitude of ${\displaystyle {\vec {c}}}$ is given by:

${\displaystyle c=|{\vec {c}}|=|{\vec {a}}||{\vec {b}}|\sin \theta =ab\sin \theta }$,

where ${\displaystyle \theta }$ is the angle between ${\displaystyle {\vec {a}}}$ and ${\displaystyle {\vec {b}}}$. The vector ${\displaystyle {\vec {a}}\times {\vec {b}}}$ is perpendicular to both ${\displaystyle {\vec {a}}}$ and ${\displaystyle {\vec {b}}}$. The direction of ${\displaystyle {\vec {a}}\times {\vec {b}}}$ is determined by a variation of the right-hand rule. If you hold your right hand as shown in the figure, your thumb is in the direction of ${\displaystyle {\vec {c}}}$, the index finger indicates the direction of ${\displaystyle {\vec {a}}}$ and your second finger points in the direction of ${\displaystyle {\vec {b}}}$. If the angle between your index and second fingers is greater than 180°, it is necessary to turn your hand upside down.

## How to calculate the cross product in vector notation

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

### Two dimensions

If
${\displaystyle {\vec {a}}=\langle a_{1},a_{2}\rangle }$
and
${\displaystyle {\vec {b}}=\langle b_{1},b_{2}\rangle }$
then
${\displaystyle {\vec {a}}\times {\vec {b}}=(a_{1}b_{2}-a_{2}b_{1}){\hat {k}}}$

or

${\displaystyle {\vec {a}}\times {\vec {b}}={\vec {c}}}$
and
${\displaystyle {\vec {c}}=\langle 0,0,a_{1}b_{2}-a_{2}b_{1}\rangle =(a_{1}b_{2}-a_{2}b_{1}){\hat {k}}}$

${\displaystyle {\hat {k}}}$ is just a symbol which says that our new vector is pointing up (in the z-direction). If you "cross" two vectors which are both in the x-y plane, the product will always be perpendicular to both vectors, and if both vectors are in the x-y plane, the only way for it to be perpendicular to both is to be in the z direction. If the value of ${\displaystyle a_{1}b_{2}-a_{2}b_{1}}$ is positive, then it points OUT of the page; if its value is negative, then it points INTO the page.

### Three dimensions

If
${\displaystyle {\vec {a}}=\langle a_{1},a_{2},a_{3}\rangle }$
and
${\displaystyle {\vec {b}}=\langle b_{1},b_{2},b_{3}\rangle }$
then
${\displaystyle {\vec {a}}\times {\vec {b}}=\langle a_{2}b_{3}-a_{3}b_{2},a_{3}b_{1}-a_{1}b_{3},a_{1}b_{2}-a_{2}b_{1}\rangle }$

## Basic properties of the cross product

${\displaystyle {\vec {a}}\times {\vec {b}}=-{\vec {b}}\times {\vec {a}}}$

${\displaystyle {\vec {a}}\times ({\vec {b}}+{\vec {c}})={\vec {a}}\times {\vec {b}}+{\vec {a}}\times {\vec {c}}}$

${\displaystyle c({\vec {a}}\times {\vec {b}})=(c{\vec {a}})\times {\vec {b}}={\vec {a}}\times (c{\vec {b}})}$