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Lagrange identity for the cross product

For three-dimensional vectors,

∣ x \times y ∣^{2} = ∣ x ∣^{2} ∣ y ∣^{2} - (x \cdot y)^{2} .

The corresponding identity in

R^{n}

is

∣ x ∣^{2} ∣ y ∣^{2} - (x \cdot y)^{2} = \frac{1}{2} \sum_{i, j} (x_{i} y_{j} - x_{j} y_{i})^{2} .

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