Cross product

The cross product is a Lie bracket on $ℝ 3$ . Given two vectors expressed in an orthonormal basis with the same orientation as the space

⃗𝐱=\vthree𝑥1𝑥2𝑥3,⃗𝐲=\vthree𝑦1𝑦2𝑦3

the cross product is given by

⃗ 𝐳 = ⃗ 𝐱 \times ⃗ 𝐲 = d e t ⎡ ⎢ ⎢ ⎢ ⎣ ˆ 𝐞 1 𝑥 1 𝑦 1 ˆ 𝐞 2 𝑥 2 𝑦 2 ˆ 𝐞 3 𝑥 3 𝑦 3 ⎤ ⎥ ⎥ ⎥ ⎦

or equivalently in index notation and using Einstein summation convention

𝑧 ℓ = 𝜖 𝑗 𝑘 ℓ 𝑥 𝑗 𝑦 𝑘

Proof of Lie bracket

#missing/proof

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