Definition of the Cross Product
The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional Euclidean space. Unlike the dot product which yields a scalar, the cross product of two vectors, say 'a' and 'b', results in a new vector 'c' that is perpendicular to both 'a' and 'b'.
Key Properties and Direction
The magnitude of the cross product vector (||a x b||) is equal to the area of the parallelogram formed by the two vectors 'a' and 'b', given by ||a|| ||b|| sin(θ), where θ is the angle between them. The direction of the resulting vector 'c' is determined by the right-hand rule: if you curl the fingers of your right hand from vector 'a' to vector 'b', your thumb points in the direction of 'c'.
Practical Applications
A classic example of the cross product's application is in physics, particularly in mechanics and electromagnetism. For instance, torque (τ), a rotational force, is calculated as the cross product of the position vector (r) from the pivot to the point where the force is applied and the force vector (F): τ = r x F. Similarly, the magnetic force (F_B) on a moving charge (q) in a magnetic field (B) is F_B = q(v x B), where v is the velocity of the charge.
Distinction from the Dot Product
It's important to differentiate the cross product from the dot product. The dot product (a · b) yields a scalar value related to the projection of one vector onto another and is maximized when vectors are parallel. Conversely, the cross product (a x b) yields a vector perpendicular to the original two and has a magnitude that is maximized when the vectors are perpendicular, becoming zero if they are parallel or anti-parallel.