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Use the Levi-Civita symbol and Kronecker delta to prove the vector identities: A x (B x C) + Bx (C x A) + Cx (A × B) = 0 ; (A x B). (C x D) = (A · C) (BD) – (A · D) (B-C) ; -

Use the Levi-Civita symbol and Kronecker delta to prove the vector identities: A x (B x C) + Bx (C x A) + Cx (A × B) = 0 ; (A x B). (C x D) = (A · C) (BD) – (A · D) (B-C) ; -

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part b only

**Exercise 18**: Use the Levi-Civita symbol and Kronecker delta to prove the vector identities: a. \( \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) + \mathbf{B} \times (\mathbf{C} \times \mathbf{A}) + \mathbf{C} \times (\mathbf{A} \times \mathbf{B}) = \emptyset \); b. \( (\mathbf{A} \times \mathbf{B}) \cdot (\mathbf{C} \times \mathbf{D}) = (\mathbf{A} \cdot \mathbf{C})(\mathbf{B} \cdot \mathbf{D}) - (\mathbf{A} \cdot \mathbf{D})(\mathbf{B} \cdot \mathbf{C}) \); c. \( (\mathbf{A} \times \mathbf{B}) \times (\mathbf{C} \times \mathbf{D}) = [(\mathbf{A} \times \mathbf{B}) \cdot \mathbf{D}] \mathbf{C} - [(\mathbf{A} \times \mathbf{B}) \cdot \mathbf{C}] \mathbf{D} \).
Transcribed Image Text:**Exercise 18**: Use the Levi-Civita symbol and Kronecker delta to prove the vector identities: a. \( \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) + \mathbf{B} \times (\mathbf{C} \times \mathbf{A}) + \mathbf{C} \times (\mathbf{A} \times \mathbf{B}) = \emptyset \); b. \( (\mathbf{A} \times \mathbf{B}) \cdot (\mathbf{C} \times \mathbf{D}) = (\mathbf{A} \cdot \mathbf{C})(\mathbf{B} \cdot \mathbf{D}) - (\mathbf{A} \cdot \mathbf{D})(\mathbf{B} \cdot \mathbf{C}) \); c. \( (\mathbf{A} \times \mathbf{B}) \times (\mathbf{C} \times \mathbf{D}) = [(\mathbf{A} \times \mathbf{B}) \cdot \mathbf{D}] \mathbf{C} - [(\mathbf{A} \times \mathbf{B}) \cdot \mathbf{C}] \mathbf{D} \).
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