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--- 183_notes:cross_product [2014/11/18 15:23] – created caballero
+++ 183_notes:cross_product [2014/11/18 15:33] (current) – caballero
@@ Line 3: / Line 3: @@
 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://en.wikipedia.org/wiki/Determinant#3.C2.A0.C3.97.C2.A03_matrices|given online]] and uses cross-multiplication. We apply that method to find the 2-by-2 determinants that can be computed.
+$$\vec{B} \times \vec{C} = det\begin{vmatrix}
+\hat{x} & \hat{y} & \hat{z} \\
+B_x & B_y & B_z \\
+C_x & C_y & C_z \\
+\end{vmatrix} = \hat{x}\begin{vmatrix}
+B_y & B_z \\
+C_y & C_z \\
+\end{vmatrix} - \hat{y} \begin{vmatrix}
+B_x & B_z \\
+C_x & C_z \\
+\end{vmatrix} + \hat{z} \begin{vmatrix}
+B_x & B_y \\
+C_x & C_y \\
+\end{vmatrix}$$
+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,
+ $\vec{B} \times \vec{C} = \langle B_yC_z - C_yB_z, C_xB_z-B_x C_z, B_xC_y - C_xB_y\rangle$