Cross Product of Vectors

Standard Definition: The cross product of two vectors $\mathbf{A}$ and $\mathbf{B}$ results in a new vector $\mathbf{C}$ that is perpendicular to both $\mathbf{A}$ and $\mathbf{B}$. The magnitude (length) of $\mathbf{C}$ is equal to the area of the parallelogram spanned by vectors $\mathbf{A}$ and $\mathbf{B}$. For two 3-dimensional vectors:

The cross product $\mathbf{A} \times \mathbf{B}$ is:

ELI5 with Real-life Example Imagine you have two sticks in your hand. You place them on a table such that they partially overlap, forming an "X" shape. The area enclosed by the two sticks represents the magnitude of the cross product. If you stand a third stick perpendicular to the table at the point where the two sticks intersect, this third stick represents the vector result of the cross product of the first two sticks. Its direction is determined by the right-hand rule: if you curl the fingers of your right hand from the first stick (vector $\mathbf{A}$) to the second stick (vector $\mathbf{B}$), your thumb points in the direction of the cross product.

Practical Applications

1. Physics: The cross product is used in calculating the torque exerted by a force. Torque is the cross product of the position vector and the force applied.
2. Computer Graphics: In 3D graphics, the cross product helps in determining the normals to surfaces which is essential for lighting calculations.
3. Aeronautics: The angular momentum of an object with respect to some point is defined as the cross product of the object's position vector and its momentum vector.