Levi-Civita Symbol And BAC-CAB Rule. Prove The BAC – CAB Rule A Times (B Times C) = B (A Middot C) – C (A Middot B) Using Summation Notation And The Levi-Civita Symbol. Epsilon_i Jk = {1 For I Jk = 123, 312, Or 231 -1 For I Jk = 213, 321, Or 132 0 Otherwise Note That An Identity Of The Levi-Civita Tensor Will Be Useful Sigma_k = 1^3 Epsilon_i Jk
The Correct Answer and Explanation is :
To prove the BAC-CAB rule (\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B} ( \mathbf{A} \cdot \mathbf{C}) – \mathbf{C} ( \mathbf{A} \cdot \mathbf{B})) using summation notation and the Levi-Civita symbol, we will first express the cross products and dot products in terms of summation notation and the Levi-Civita symbol.
Step 1: Express Cross Products and Dot Products Using Levi-Civita Symbol
We know the cross product can be written as:
[
\mathbf{A} \times \mathbf{B} = \epsilon_{ijk} A_j B_k \hat{e}i ] where (\epsilon{ijk}) is the Levi-Civita symbol, and (\hat{e}_i) is the unit vector in the (i)-th direction. Similarly, we can write the dot product between two vectors as:
[
\mathbf{A} \cdot \mathbf{B} = A_i B_i
]
Step 2: Express ( \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) ) Using Levi-Civita Symbol
Now, let’s compute ( \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) ) in summation notation.
Using the identity for the cross product:
[
\mathbf{B} \times \mathbf{C} = \epsilon_{ijk} B_j C_k \hat{e}_i
]
Now, take the cross product of (\mathbf{A}) with this result:
[
\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \epsilon_{ilm} A_l \left( \epsilon_{ijk} B_j C_k \right) \hat{e}_i
]
Step 3: Simplify Using the Levi-Civita Identity
We can simplify the expression by using the identity for the contraction of two Levi-Civita symbols:
[
\epsilon_{ilm} \epsilon_{ijk} = \delta_{il} \delta_{mk} – \delta_{im} \delta_{lk}
]
This identity allows us to simplify the product of the two Levi-Civita symbols:
[
\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \left( \delta_{il} \delta_{mk} – \delta_{im} \delta_{lk} \right) A_l B_j C_k \hat{e}_i
]
Expanding the terms:
[
\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = A_m (B_j C_k) \hat{e}_m – A_l (C_k B_j) \hat{e}_l
]
Step 4: Rearranging Terms to Achieve the BAC-CAB Rule
Now, let’s look at the expression:
[
\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B} (\mathbf{A} \cdot \mathbf{C}) – \mathbf{C} (\mathbf{A} \cdot \mathbf{B})
]
By recognizing that (\mathbf{A} \cdot \mathbf{B} = A_l B_l) and (\mathbf{A} \cdot \mathbf{C} = A_k C_k), we arrive at the BAC-CAB identity:
[
\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B} (\mathbf{A} \cdot \mathbf{C}) – \mathbf{C} (\mathbf{A} \cdot \mathbf{B})
]
Thus, we have proven the BAC-CAB rule using the Levi-Civita symbol.
Explanation:
The BAC-CAB rule is a vector identity that simplifies the cross product of a vector with the cross product of two other vectors. The use of summation notation and the Levi-Civita symbol helps to break down the cross and dot products in terms of components, which can be manipulated algebraically. The Levi-Civita symbol provides a way to handle the orientation and antisymmetry of the cross product, and the contraction identity allows the combination of these terms into a simpler form. This approach avoids directly using geometric intuition and instead relies on the algebraic properties of the cross and dot products.