Vector cross product

The vector cross product, also known as the vector product or cross product, is an operation performed on two vectors in three-dimensional space that results in a new vector that is perpendicular to both of the original vectors. The cross-product is defined as:

A × B = |A| |B| sin(θ) n

where:

1. A and B are the two vectors being multiplied
2. |A| and |B| are the magnitudes of A and B, respectively
3. θ is the angle between A and B, measured in radians
4. n is a unit vector perpendicular to both A and B in the direction determined by the right-hand rule

The right-hand rule states that if you point your right thumb in the direction of A and your right index finger in the direction of B, the vector product will point in the direction of your middle finger.

The magnitude of the cross product is given by:

|A × B| = |A| |B| sin(θ)

To calculate the components of the cross product vector, you can use the following determinant:

A × B = | i j k |
| Ax Ay Az |
| Bx By Bz |

where i, j, and k are the unit vectors in the x, y, and z directions, respectively, and Ax, Ay, Az, Bx, By, and Bz are the components of vectors A and B.

The resulting vector will have components (Cx, Cy, Cz) given by:

Cx = AyBz – AzBy
Cy = AzBx – AxBz
Cz = AxBy – AyBx

The vector cross product is used in various applications, including physics, engineering, and computer graphics, to calculate torque, angular momentum, and the normal vector to a plane, among other things.