# Cross Products

• Apr 26th 2010, 10:21 AM
Diemo
Cross Products
Ok, so I was talking to my lecturer about how the Electric field and the Magneticf field are related in Dielectrics. Basically,
$\displaystyle \vec{B}=\sqrt{\mu\epsilon}\frac{\vec{k}\times\vec{ E}}{|k|}$
Now, I thought about this to get a value for B from this, and thought that what you should do is ust assume everything is unit vectors to start with.
Then, if you take the cross product of the whole thing with respect to K, you get

$\displaystyle k\times B\times k=\frac{1}{\sqrt{\mu\epsilon}}E\cross k =B$. However, my lecturer says that I am missing a minus, and that two cross products introduce a minus sign. Can someone show me why this is please, as when I use the unit vector $\displaystyle \vec{i},\vec{j},\vec{k}$ I don't get a minus sign.
• Apr 26th 2010, 04:29 PM
dwsmith
Quote:

Originally Posted by Diemo
Ok, so I was talking to my lecturer about how the Electric field and the Magneticf field are related in Dielectrics. Basically,
$\displaystyle \vec{B}=\sqrt{\mu\epsilon}\frac{\vec{k}\times\vec{ E}}{|k|}$
Now, I thought about this to get a value for B from this, and thought that what you should do is ust assume everything is unit vectors to start with.
Then, if you take the cross product of the whole thing with respect to K, you get

$\displaystyle k\times B\times k=\frac{1}{\sqrt{\mu\epsilon}}E\cross k =B$. However, my lecturer says that I am missing a minus, and that two cross products introduce a minus sign. Can someone show me why this is please, as when I use the unit vector $\displaystyle \vec{i},\vec{j},\vec{k}$ I don't get a minus sign.

I think this is what you are looking for not sure though.

$\displaystyle \begin{bmatrix} + & - & +\\ - & + & -\\ + & - & + \end{bmatrix}$ so when you do your cross product you have, expansion across row 1, $\displaystyle (+1)*a_{11}M_{11}\mathbf{i}+(-1)*a_{12}M_{12}\mathbf{j}+(+1)*a_{13}M_{13}\mathbf {k}$