PlanetPhysics/Cross Product

The cross product or vector product is defined by

$$ \mathbf{A} \times \mathbf{B} = \left ( A_y B_z - A_z B_y \right ) \mathbf{\hat{i}} + \left ( A_z B_x - A_x B_z \right ) \mathbf{\hat{j}} + \left ( A_x B_y - A_y B_x \right ) \mathbf{\hat{k}}$$

Like the dot product, it is useful to look at its geometric definition and properties. Instead of the cosine of the angle between the two vectors the cross product is defined geometrically as

$$ \mathbf{A} \times \mathbf{B} = \left | \mathbf{A} \right | \left | \mathbf{B} \right | \sin \theta \mathbf{\hat{n}} $$

It is important to see that the unit vector $$\mathbf{\hat{n}}$$ is normal to the plane defined by the two vectors with the direction determined by the right hand rule.

It can be easier to remember the definition of the cross product with the determinant formulation

$$ \mathbf{A} \times \mathbf{B} = \left| \begin{matrix} \mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{matrix}\right| = \left ( A_y B_z - A_z B_y \right ) \mathbf{\hat{i}} + \left ( A_z B_x - A_x B_z \right ) \mathbf{\hat{j}} + \left ( A_x B_y - A_y B_x \right ) \mathbf{\hat{k}}$$