 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2018/2/a/solution

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

, expressed through the Kronecker delta, gives

(a \times (b \times c))_{i} = ϵ_{ijk} a_{j} ϵ_{k l m} b_{l} c_{m} = b_{i} a_{j} c_{j} - c_{i} a_{j} b_{j} .

The last two contractions are dot products, proving the vector triple product identity

a \times (b \times c) = (a \cdot c) b - (a \cdot b) c

 New to topics? Read the docs here!