Vector Analysis | Quantum Science Philippines
Quantum Science Philippines

Vector Analysis

Prove:

\vec{a}\cdot(\vec{b}\times\vec{c})=\vec{b}\cdot(\vec{c}\times\vec{a})=\vec{c}\cdot(\vec{a}\times\vec{b})

where:

\vec{a}=a_i\widehat{e_i}

 

\vec{b}=b_j\widehat{e_j}

 

\vec{c}=c_k\widehat{e_k}

Sol’n:

\vec{b}\times\vec{c}=b_jc_k(\widehat{e_j}\times\widehat{e_k})

 

\vec{b}\times\vec{c}=b_jc_k\in_{jkl}\widehat{e_l}

then:

\vec{a}\cdot(\vec{b}\times\vec{c})=a_i\widehat{e_i}\cdot b_jc_k\in_{jkl}\widehat{e_l}

 

=a_ib_jc_k\in_{jkl}\widehat{e_i}\cdot\widehat{e_l}

 

=a_ib_jc_k\in_{jkl}\delta_{il}

 

=a_ib_jc_k\in_{jki}

 

=b_jc_ka_i\in_{jki}

 

=b_j\widehat{e_j}\cdot c_ka_i\in_{kij}\widehat{e_j}

 

\vec{a}\cdot(\vec{b}\times\vec{c})=\vec{b}\cdot(\vec{c}\times\vec{a})

or

=c_k a_ib_j\in_{ijk}

 

=c_k\widehat{e_k}\cdot a_ib_j\in_{ijk}\widehat{e_k}

 

\vec{a}\cdot(\vec{b}\times\vec{c})=\vec{c}\cdot(\vec{a}\times\vec{b})

 

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