Question

Use the $\varepsilon_{\text {ijk }}$ notation and derive the identity $$(\mathbf{A} \times \mathbf{B}) \times(\mathbf{C} \times \mathbf{D})=(\mathbf{A B D}) \mathbf{C}-(\mathbf{A B C}) \mathbf{D}$$

          Use the $\varepsilon_{\text {ijk }}$ notation and derive the identity
$$(\mathbf{A} \times \mathbf{B}) \times(\mathbf{C} \times \mathbf{D})=(\mathbf{A B D}) \mathbf{C}-(\mathbf{A B C}) \mathbf{D}$$

Added by Lauren W.

Instant Answer

Solved by Expert Aarti Kumari

Step 1

Now, let's compute the cross product of these two vectors: $$(\mathbf{A} \times \mathbf{B}) \times (\mathbf{C} \times \mathbf{D}) = (\varepsilon_{ijk} A_j B_k \mathbf{e}_i) \times (\varepsilon_{lmn} C_m D_n \mathbf{e}_l)$$ Using the cross product properties, we Show more…

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Use the $\varepsilon_{\text {ijk }}$ notation and derive the identity $$(\mathbf{A} \times \mathbf{B}) \times(\mathbf{C} \times \mathbf{D})=(\mathbf{A B D}) \mathbf{C}-(\mathbf{A B C}) \mathbf{D}$$