Cross product by epsilon ijk method

In summary, the problem given is to calculate Ax(BxC) using the epsilon ijk rule and kronecker delta equation. The Levi-Civita symbol is used to collapse products of the symbol and the solution can be found by applying the relations on the page provided.
  • #1
physiker99
36
0

Homework Statement


I need to calculate Ax(BxC) -A, B, C are vectors, apart from bac-cab rule, I need to get it by epsilon ijk -Following I will note it by e(ijk).

Homework Equations


I expect answerer to know kronecker delta equation and vector multiplications.


The Attempt at a Solution



e(ijk)*A(i)*(B*C)(j)
= e(ijk)*A(i)

(BxC)(j) = e(mjn)B(n)C(m)

After that part I'm stuck. As you can see below, I rotate the ijk's and mjn's to match j's at the end, but I fail.

e(ijk)*A(k)*e(mjn)*B(n)*C(m)

Can you help me after that?
Lettters in parantheses (i, j, k, m, n) are supposed to indicate the indices.
 
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  • #2
  • #3
I know Wikipedia, but if someone would bother to actually solve the question, I will be glad.
 
  • #4
physiker99 said:
I know Wikipedia, but if someone would bother to actually solve the question, I will be glad.

I'm afraid that's not how things work around here-- if you come here asking a question you should be prepared to put in the work to find the solution. Homework helpers are here to help guide you to a solution.

So, can you apply some of the relations on the page vela showed you to your problem?
 

FAQ: Cross product by epsilon ijk method

What is the cross product by epsilon ijk method?

The cross product by epsilon ijk method is a mathematical operation used to find a vector that is perpendicular to two given vectors in three-dimensional space. It involves using the Levi-Civita symbol (also known as the permutation symbol) to calculate the components of the resulting vector.

How is the cross product by epsilon ijk method calculated?

To calculate the cross product using the epsilon ijk method, you first write out the components of the two given vectors in a determinant form. Then, you replace the i, j, and k values with the corresponding components of the Levi-Civita symbol. Next, you expand the determinant and simplify the resulting expression to get the components of the resulting vector.

What is the significance of the epsilon ijk method in vector algebra?

The epsilon ijk method is important in vector algebra because it allows us to find a vector that is perpendicular to two given vectors. This is useful in various applications, such as calculating torque, determining the direction of magnetic fields, and solving problems in mechanics and electromagnetism.

Can the cross product by epsilon ijk method be used in higher dimensions?

No, the cross product by epsilon ijk method is only defined in three-dimensional space. In higher dimensions, the concept of perpendicularity becomes more complex, and the cross product is replaced by the wedge product or exterior product.

Are there any properties of the cross product by epsilon ijk method?

Yes, there are several properties of the cross product by epsilon ijk method. Some of these include the distributive property, the fact that the resulting vector is perpendicular to the two given vectors, and the right-hand rule, which determines the direction of the resulting vector. It is also anti-commutative, meaning that the order of the vectors matters when calculating the cross product.

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