Tensor Notation for Triple Scalar Product Squared

In summary: Yes, that's correct. The order of operations should be followed when doing the dot product: first, you compute the cross product of the two vectors.
  • #1
203
0

Homework Statement



Hi all,

Here's the problem:

Prove, in tensor notation, that the triple scalar product of (A x B), (B x C), and (C x A), is equal to the square of the triple scalar product of A, B, and C.

Homework Equations


The Attempt at a Solution



I started by looking at the triple scalar product of A, B, and C. I know that I can write that as ε[itex]_{ijk}[/itex]A[itex]_{i}[/itex]B[itex]_{j}[/itex]C[itex]_{k}[/itex]. This is about as far as I got.

Initially, I did something different that got me further, but I think I was interpreting the problem incorrectly. What I did was set a = (AxB), b = (BxC), and c = (CxA), thinking that the square of the triple scalar product was referring to the triple scalar product of a, b, and c, but upon reading the problem again, I don't think that that's right.

That being said, I'm not sure where to go from here.

I know that the subject of tensors is a particularly difficult one to discuss over the web, especially without just spelling out the answer -- which I'm certainly not looking for. But that being said, I'm pretty lost, and could use a lot of help in the right direction.

Thanks so much!
 
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  • #2
forestmine said:

Homework Statement



Hi all,

Here's the problem:

Prove, in tensor notation, that the triple scalar product of (A x B), (B x C), and (C x A), is equal to the square of the triple scalar product of A, B, and C.

Homework Equations





The Attempt at a Solution



I started by looking at the triple scalar product of A, B, and C. I know that I can write that as ε[itex]_{ijk}[/itex]A[itex]_{i}[/itex]B[itex]_{j}[/itex]C[itex]_{k}[/itex]. This is about as far as I got.

Initially, I did something different that got me further, but I think I was interpreting the problem incorrectly. What I did was set a = (AxB), b = (BxC), and c = (CxA), thinking that the square of the triple scalar product was referring to the triple scalar product of a, b, and c, but upon reading the problem again, I don't think that that's right.

That being said, I'm not sure where to go from here.

I know that the subject of tensors is a particularly difficult one to discuss over the web, especially without just spelling out the answer -- which I'm certainly not looking for. But that being said, I'm pretty lost, and could use a lot of help in the right direction.

Thanks so much!

Knowing that [itex]\vec{A}\cdot(\vec{B}\times\vec{C}) = \epsilon_{ijk}A_{i}B_{j}C_{k}[/itex] is an excellent start.

The next thing I would try to figure out is how to express the cross product between two cross products [itex](\vec{A}\times \vec{B})\times(\vec{B}\times\vec{C})[/itex]

The following identity should be useful (note that the k index is repeated, and so summed over)
[itex]\epsilon_{ijk}\epsilon_{k\ell m} = \delta_{i\ell}\delta_{jm}-\delta_{im}\delta_{j\ell}[/itex]
With this identity, you can show that
[itex]\vec{A}\times (\vec{B}\times\vec{C}) =(\vec{A}\cdot\vec{C})\vec{B}-(\vec{A}\cdot\vec{B})\vec{C}[/itex]
among other vector identities.
 
  • #3
Thanks for the response!

Ok, well, I know that (AxB)[itex]_{i}[/itex] = ε[itex]_{ijk}[/itex]A[itex]_{j}[/itex]B[itex]_{k}[/itex]. And I can say that, (BxC)[itex]_{i}[/itex] = ε[itex]_{ilm}[/itex]B[itex]_{l}[/itex]C[itex]_{m}[/itex].

If I write that all as one term,
ε[itex]_{ijk}[/itex]A[itex]_{j}[/itex]B[itex]_{k}[/itex]ε[itex]_{ilm}[/itex]B[itex]_{l}[/itex]C[itex]_{m}[/itex]

then that equals,

(δ[itex]_{jl}[/itex]δ[itex]_{km}[/itex] - δ[itex]_{jm}[/itex]δ[itex]_{kl}[/itex])A[itex]_{j}[/itex]B[itex]_{k}[/itex]B[itex]_{l}[/itex]C[itex]_{m}[/itex]

and I know that I can do some further simplification of that, but I'm worried about simplifying it too much before incorporating yet the third cross product. Am I on the right track at least?
 
  • #4
Yes this is exactly on the right track. The formula you give would be for the dot product of the two cross products, i.e., [itex](\vec{A}\times\vec{B})\cdot(\vec{B}\times\vec{C})[/itex]. If each vector in the second cross product is expressed as a cross product of two other vectors, you should be able to get a final result.
 
  • #5
jfizzix said:
Yes this is exactly on the right track. The formula you give would be for the dot product of the two cross products, i.e., [itex](\vec{A}\times\vec{B})\cdot(\vec{B}\times\vec{C})[/itex]. If each vector in the second cross product is expressed as a cross product of two other vectors, you should be able to get a final result.

Hm, ok, I'm not sure I follow here. If I computed the dot product of two cross products, isn't that wrong, in that I should first compute the cross product of (BxC) x (CxA), before taking the dot product that I did in the step before?
 
  • #6
The expression you started with, ##(\varepsilon_{ijk}A_j B_k)(\varepsilon_{ilm}B_l C_m)##, is equal to ##(\vec{A}\times\vec{B})_i (\vec{B}\times\vec{C})_i##, which is equal to ##(\vec{A}\times\vec{B})\cdot(\vec{B}\times\vec{C})##. That's not what you want, right? So don't combine the two cross products that way.

Go back to your previous attempt where you said
\begin{align*}
\vec{a} &= \vec{A}\times\vec{B} \\
\vec{b} &= \vec{B}\times\vec{C} \\
\vec{c} &= \vec{C}\times\vec{A}
\end{align*} The triple scalar product of those three vectors is ##\varepsilon_{ijk}a_i b_j c_k##. Now you want to substitute in expressions for ##a_i## in terms of the components of ##\vec{A}## and ##\vec{B}##, and so on, and the simplify it.
 

1. What is Tensor Notation for Triple Scalar Product Squared?

Tensor Notation for Triple Scalar Product Squared is a mathematical notation used to represent the square of the triple scalar product between three vectors. It is commonly used in physics and engineering to simplify and streamline calculations involving vector quantities.

2. How is Tensor Notation for Triple Scalar Product Squared written?

In Tensor Notation for Triple Scalar Product Squared, the square of the triple scalar product is represented by the notation (A x B x C)^2, where A, B, and C are three vectors. The square of the triple scalar product is also equal to the determinant of the matrix formed by the three vectors.

3. What is the significance of using Tensor Notation for Triple Scalar Product Squared?

Tensor Notation for Triple Scalar Product Squared allows for a more concise and elegant representation of the square of the triple scalar product. It also simplifies calculations and makes them more easily applicable to higher dimensions.

4. Can Tensor Notation for Triple Scalar Product Squared be applied to more than three vectors?

Yes, Tensor Notation for Triple Scalar Product Squared can be extended to any number of vectors. The notation (A1 x A2 x A3 x ... x An)^2 is used to represent the square of the n-fold scalar product between n vectors.

5. How is Tensor Notation for Triple Scalar Product Squared used in real-world applications?

Tensor Notation for Triple Scalar Product Squared is commonly used in fields such as mechanics, electromagnetism, and fluid dynamics to simplify and generalize vector calculations. It also has applications in computer graphics and machine learning for representing and manipulating multi-dimensional data.

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