The cross or vector product of two vectors A and B is defined as:
n  unit vector whose direction is perpendicular to vectors A and B.
Note: the direction of A ✕ B is normal to the plane defined by A and B and
is pointing according to the righthand screw rule.
From the definition of the cross product the following relations between the vectors are apparent:
The vector product is written as:
This expression may be written as a determinant: 

The cross product obeys the following laws:
The commutative law does not hold for cross product because:
A ✕ B = − (B ✕ A) and A ✕ (B ✕ C) ≠ (A ✕ B) ✕ C
Distributive law: 
A ✕ (B + C) = A ✕ B + A ✕ C 

(A + B) ✕ C = (A ✕ C) + (B ✕ C) 
If m is a scalar, then: 
m(A ✕ B) = (mA) ✕ B = A ✕ (mB) = (A ✕ B)m 
Triple scalar product is defined as the determinant:
(A ✕ B) ✕ C = C ✕ (A ✕ B) = C ✕ (B ✕ A)
Other relations: 
A · (A ✕ C) = 0 

A ✕ (B ✕ C) + B ✕ (C ✕ A) + C ✕ (A ✕ B) = 0 

A ✕ (B ✕ C) =(A · C)B − (A · B)C 

(A ✕ B) ✕ C = (A · C)B − (B · C)A 

(A ✕ B) · (C ✕ D) = (A · C)(B · D) − (A · D)(B · C) 
