 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2021/8/a/ii/solution

Contract the Levi-Civita symbol identity

ε_{ijk} ε_{i ℓ m} = δ_{j ℓ} δ_{km} - δ_{jm} δ_{k ℓ}

∣ a \times b ∣^{2} = ε_{ijk} a_{j} b_{k} ε_{i ℓ m} a_{ℓ} b_{m} = (δ_{j ℓ} δ_{km} - δ_{jm} δ_{k ℓ}) a_{j} b_{k} a_{ℓ} b_{m} = ∣ a ∣^{2} ∣ b ∣^{2} - (a \cdot b)^{2} .

This Lagrange identity for the cross product includes zero and parallel vectors without a separate division by their norms.

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