Introduction:

The cross product of two vectors is an operation in vector calculus and physics. It results in a new vector perpendicular to the plane containing the original vectors.



Understanding the Cross Product:



The cross product applies to vectors in three-dimensional space.

It yields a vector orthogonal to both of the original vectors.

Calculating the Cross Product:



The magnitude of the cross product is given by the formula: |A × B| = |A| |B| sin(θ), where:

|A × B| is the magnitude of the cross product.

|A| and |B| are the magnitudes of the original vectors.

θ is the angle between vectors A and B.

The direction of the cross product is determined by the right-hand rule.

The Right-Hand Rule:



Point the index finger of your right hand in the direction of the first vector (A).

Point your middle finger in the direction of the second vector (B).

Your thumb then points in the direction of the cross product (A × B).

Example:



For vectors A = (a1, a2, a3) and B = (b1, b2, b3), the cross product A × B is a vector given by:

(a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1).

Applications:



The cross product is used in physics to calculate torque and angular momentum.

In engineering, it's used for finding normal vectors and in determining area and volume.