To take the cross product of two vectors ($\vec{B} \times \vec{C}$) in Cartesian coordinates in general, we set up a special 3-by-3 matrix that has as its rows the Cartesian unit vectors ($\hat{x}$, $\hat{y}$, and $\hat{z}$), the components of the first vector ($B_x$, $B_y$, and $B_z$), and the components of the second vector ($C_x$, $C_y$, and $C_z$). The columns are organized by component. The determinant of this matrix will give us the cross product of the two vectors:

$$\vec{B} \times \vec{C}$$