CROSS PRODUCT FORMULA

Cross product (or vector product) is a type of vector multiplication and cross product of two vectors is denoted by \(\vec { A } \times \vec { B }\). The result of a cross product is vector and shown with cross product formula below.

\(\vec { A } \times \vec { B } =\vec { C }\)
Cross products are used when dealing with torques and angular momentums. There are two methods for the calculation of cross product.

Calculation of Cross Product by Using Components

The components of the cross product can be calculated with the following procedure and cross product formula if the components of \(\vec { A } \) and \(\vec { B }\) vectors are known. This method is also used to find determinant of a 3×3 matrix.

 

Cross Product (Vector Product) of Vectors

\(\vec { C } =({ A }_{ y }{ B }_{ z }-{ A }_{ z }{ B }_{ y })\hat { x } +({ A }_{ z }{ B }_{ x }-{ A }_{ x }{ B }_{ z })\hat { y } +({ A }_{ x }{ B }_{ y }-{ A }_{ y }{ B }_{ x })\hat { z }  \)

 

Calculation of Cross Product by Geometry

Second method for the calculation of cross product is a geometrical method as shown below. The two vectors are so drawn that the tails of the vectors are at same point. The resultant vector of the multiplication is perpendicular to both of vector \(\vec { A } \) and \(\vec { B }\).

\(\vec { C } =\vec { A } \times \vec { B } =\left| A \right| \left| B \right| \sin { \theta } \)

Cross Product Formula

If the angle between two vectors (θ) is 0° or 180° (parallel or antiparallel vectors), the cross product is zero.  It’s immediately seen from the cross product formula that the result of cross product can be larger than 0, smaller than 0 or equal to 0. In particular, the vector product of any vector with itself is zero.

Right Hand Rule for Cross Product

There are always two directions perpendicular to a given plane so the direction of the vector results from the cross (vector) product of vectors is found as follows.

The first vector of the multiplication is taken \(\vec { A } \) (for the case above) and rotated over the shortest possible angle to \(\vec { B } \). If you curl your right hand fingers around the perpendicular line to \(\vec { A } \) and \(\vec { B }\)  so that your fingertips point in the direction of rotation; your thumb will then point in the direction of resultant vector and this is called right-hand rule. Right handed coordinate system shall be used for right hand rule.

Right Hand Rule for Vector Product

The direction of cross product is decided with right hand rule described above. If the resultant vector is pointing out of plane direction, the vector is indicated with a circle and a dot. If the resultant vector is pointing into the plane, the tail of the arrow will be seen and it’s indicated with a cross.

So for \(\vec { B } \times \vec { A }\), the magnitude will be same with \(\vec { A } \times \vec { B }\) but the direction of the result will towards out of plane.

\(\vec { A } \times \vec { B }= -\vec { B } \times \vec { A }\)

Vector Product Direction out of the Plane

Cross Product Example

What is the cross product of \(\vec { A } \) and \(\vec { B }\)  ?

\(\vec { A } =\hat { x }\)
\(\vec { B } =\hat { y }\)
\({ A }_{ x }=1,{ B }_{ y }=1\Longrightarrow \)
\(\vec { A } \times \vec { B } = ({ A }_{ y }{ B }_{ z }-{ A }_{ z }{ B }_{ y })\hat { x } + ({ A }_{ z }{ B }_{ x }-{ A }_{ x }{ B }_{ z })\hat { y }+ ({ A }_{ x }{ B }_{ y }-{ A }_{ y }{ B }_{ x })\hat { z }\)
\(\vec { A } \times \vec { B } =\hat { z }\)

 

Additional Resources

 

Reference