The cross product of two vectors $A$ and $B$ in 3D space is a vector $C$ that is perpendicular to both $A$ and $B$. It is denoted by $A \times B$.

Determinant Formula:

$$ A \times B = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} $$

Expanding the determinant gives:

$$ A \times B = (A_yB_z - A_zB_y)\mathbf{i} - (A_xB_z - A_zB_x)\mathbf{j} + (A_xB_y - A_yB_x)\mathbf{k} $$

Magnitude:

The magnitude of the cross product, $\|A \times B\|$, is equal to the area of the parallelogram formed by vectors $A$ and $B$.

$$ \|A \times B\| = \|A\| \|B\| \sin(\theta) $$