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#### skatenerd

##### Active member

- Oct 3, 2012

- 114

For example, I have a problem where I am to prove

$$\vec{A}\bullet\vec{B}\times\vec{C} = \vec{B}\bullet\vec{C}\times\vec{A} = \vec{C}\bullet\vec{A}\times\vec{B}$$

using tensor notation to avoid having to write out all the terms.

So I know the very left side of this equation would look like

$$\vec{A}\bullet(\vec{B}\times\vec{C}) = A_i (\vec{B}\times\vec{C})_i = A_i \varepsilon_{ijk} B_j C_k$$

But then I get confused when trying to assign the indices for the next two parts of the equation.

Would the second part look like this:

$$\vec{B}\bullet(\vec{C}\times\vec{A}) = B_j (\vec{C}\times\vec{A})_j = B_j \varepsilon_{jkl} C_k A_i$$

Or would the indices of the epsilon be the same as for the first part (\(\varepsilon_{ijk}\))?

Same confusion goes for the first part. The reason I have this uncertainty in my mind is because I know with the triple vector product, you have to introduce 2 extra indices. So I guess my lack of complete understanding of these functions is leaving me confused with my problem. Thanks in advance for any guidance.