x \times y

x

and

y

, of magnitude

∣ x ∣∣ y ∣ sin α

(x \times y)_{i} = ε_{ijk} x_{j} y_{k},

where

ε_{ijk}

is the Levi-Civita symbol. Therefore

[x \times (x \times y)]_{i} = ε_{ijk} x_{j} ε_{k l m} x_{l} y_{m} .

Using

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

gives

[x \times (x \times y)]_{i} = x_{i} x_{j} y_{j} - y_{i} x_{j} x_{j} .

x \times (x \times y) = x (x \cdot y) - y (x \cdot x) .

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