Cross Product Formula With Solved Examples and Properties
In this article, you will learn what the cross product of two vectors is and how it is calculated.
What is Cross Product?
In vector analysis, the cross product is a multiplicative product of two vectors in three-dimensional space which results in a vector perpendicular to both vectors. It is denoted by a×b. It is a binary operation on two vectors.
The cross product is an important key point in physics also, because it is used to express the graphical representation of two quantities and their directions.
Cross Product Formula
The cross product is also known as a vector product because it is the product of two vectors. If A and B are two independent vectors, their cross product formula can be written as:
The resulting vector will be perpendicular to both vectors A and B.
Since the cross product of two vectors results in another vector so we can write that:
$$ \vec A ×\vec B \;=\;\vec C $$
Where C denotes the resultant of A and B vectors and the arrow head is the sign of vector representation.
How to Find a Cross Product?
If A and B are two vectors such as:
$$ \vec A \;=\; ai + bj + ck \;\;and\;\; \vec B \;=\; di + ej + fk $$
The cross product of two vectors A and B can be calculated by using the following formula.
$$ \vec A ×\vec B \;=\;\vec C $$
Where, the resultant vector is calculated by using matrix notation.
$$ \vec C \;=\; \begin{vmatrix} i & j & k \\ a & b & c \\ d & e & f \\ \end{vmatrix} $$ $$ \vec C \;=\; i \begin{vmatrix} b & c \\ e & f \\ \end{vmatrix} \;-\; j \begin{vmatrix} a & c \\ d & f \\ \end{vmatrix} \;+\; k \begin{vmatrix} a & b \\ d & e \\ \end{vmatrix} $$
See the below example to understand how to calculate the cross product of two vectors.
Properties of Cross Product
There are some properties of cross product on which it is based. These are:
A × B × C = (A × B) + (A × C)
A × 0 = 0
A × A = 0
- Anti-commutative property:
A × B = -B × A
- Distributive property:
- Cross product of the zero vector:
- Cross product of the vector with itself:
FAQ’s
How do you Find the Cross Product?
The cross product of two vectors can be calculated by using right hand rule which is:
Grab the fingers of the right hand in such a way that the two fingers represent the direction of two vectors and the thumb will represent the direction of the resultant vector which is perpendicular to the original vectors.
Is Cross Product Commutative?
No, the cross product is not commutative. It is anti-commutative such as:
A × B = -B × A
