Classical Mechanics (Kibble & Berkshire) · Problem 1.11

Velocity, acceleration and scalar angular velocity — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · vector-algebra-identities

Problem

Prove the identities (i)

(\boldsymbol{a}\times\boldsymbol{b}) \cdot (\boldsymbol{c}\times\boldsymbol{d}) = (\boldsymbol{a} \cdot \boldsymbol{c})(\boldsymbol{b} \cdot \boldsymbol{d}) - (\boldsymbol{a} \cdot \boldsymbol{d})(\boldsymbol{b} \cdot \boldsymbol{c})

(ii)

(\boldsymbol{a}\times\boldsymbol{b})\times(\boldsymbol{c}\times\boldsymbol{d}) = [\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{d}]\,\boldsymbol{c} - [\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}]\,\boldsymbol{d}

(iii)

\boldsymbol{a}\times(\boldsymbol{b}\times\boldsymbol{c}) + \boldsymbol{c}\times(\boldsymbol{a}\times\boldsymbol{b}) + \boldsymbol{b}\times(\boldsymbol{c}\times\boldsymbol{a}) = \boldsymbol{0}

(Jacobi's identity)