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| 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: | 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: | ||
| - | $$\vec{B} \times \vec{C}$$ | + | $$\vec{B} \times \vec{C} = det\begin{vmatrix} |
| + | \hat{x} & \hat{y} & \hat{z} \\ | ||
| + | B_x & B_y & B_z \\ | ||
| + | C_x & C_y & C_z \\ | ||
| + | \end{vmatrix}$$ | ||
| + | A useful way to remember how to take the determinant is [[http:// | ||
| + | $$\vec{B} \times \vec{C} = det\begin{vmatrix} | ||
| + | \hat{x} & \hat{y} & \hat{z} \\ | ||
| + | B_x & B_y & B_z \\ | ||
| + | C_x & C_y & C_z \\ | ||
| + | \end{vmatrix} = \hat{x}\begin{vmatrix} | ||
| + | B_y & B_z \\ | ||
| + | C_y & C_z \\ | ||
| + | \end{vmatrix} - \hat{y} \begin{vmatrix} | ||
| + | B_x & B_z \\ | ||
| + | C_x & C_z \\ | ||
| + | \end{vmatrix} + \hat{z} \begin{vmatrix} | ||
| + | B_x & B_y \\ | ||
| + | C_x & C_y \\ | ||
| + | \end{vmatrix}$$ | ||
| + | |||
| + | We further cross-multiply to find the determinants of the 2-by-2 matrices, | ||
| + | |||
| + | $$\vec{B} \times \vec{C} = \hat{x}\left(B_yC_z - C_yB_z\right)- \hat{y} \left(B_x C_z - C_xB_z\right) + \hat{z} \left(B_xC_y - C_xB_y\right)$$ | ||
| + | |||
| + | So, in general, the cross product in Cartesian coordinates is given by, | ||
| + | |||
| + | $$\vec{B} \times \vec{C} = \langle B_yC_z - C_yB_z, C_xB_z-B_x C_z, B_xC_y - C_xB_y\rangle$$ | ||