Cross product

The cross product is a mathematical operation which can be done between two three-dimensional vectors. It is often represented by the symbol formula_1. After performing the cross product, a new vector is formed. The cross product of two vectors is always perpendicular (it makes a corner-shaped angle) to both of the vectors which were "crossed". This means that cross product normally works only in three-dimensional space; that is, anywhere where you can go up or down, left or right, and forward or backwards.
Information about vectors.
A vector is information that describes two things: which way (direction), and how much (magnitude). Usually, the simplest vector can be an arrow. How long is the line, and which way does the line and the pointed end of the arrow go?
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 (pushing or pulling in a circle, like twisting off a bottle cap.) or moment. Another important application involves the magnetic field.
Visualizing the cross product in three dimensions.
The cross product of formula_2 and formula_3 is a vector that we shall call formula_4:
The vector formula_6 is perpendicular to both formula_7 and formula_8. The direction of formula_6 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 formula_4 (the cross product of formula_7 and formula_8), with the index finger pointing in the direction that formula_7 points, and the middle finger pointing in the direction that formula_8 points. If the angle between the index (your pointer finger) and middle (second finger away from your thumb) fingers is greater than 180°, then you need to turn the 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.
Since cross products usually only work for three-dimensional vectors, the calculation of cross product in two dimensions thinks of the vectors as if they are on a flat surface (like a sheet of paper) in three dimensions (like a room or a large cube).
More specifically, if
formula_15
and
formula_16
then
formula_17
or
formula_18
and
formula_19
where formula_20 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 formula_21 is positive, then it points out of the page; if its value is negative, then it points into the page.
Three dimensions.
There are two ways to find the cross product of two 3D vectors: with coordinate notation or with angle.
Coordinate notation.
Given vectors formula_7 and formula_8, where 
formula_24
and 
formula_25
Then the cross product of formula_7 and formula_8 is: 
formula_28.
With angle.
Given vectors formula_7 and formula_8, where 
formula_24
and 
formula_25
Then the cross product of formula_7 and formula_8 is: 
formula_35,
where formula_36 is the angle between formula_7 and formula_8, ‖a‖ and ‖b‖ are the magnitudes of vectors formula_2 and formula_3, and n is a unit vector perpendicular to the plane containing formula_2 and formula_3.