The cross product is a mathematical operation which can be done between two three-dimensionalvectors. It is often represented by the symbol . 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 cross product is normally only valid in three-dimensional space.
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.
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.