I don't understand some of the terminology in this question. I googled reciprocal vectors and got an article on reciprocal lattices, but I'm not sure if that is what they are talking about in this question. Also, when they say that ${\bf A}$, ${\bf B}$, and ${\bf C}$ are defined by … plus cyclic permutations, again I looked at the wikipedia article on the subject, but I still do not understand the concept. Does anyone have a link for a clear explanation?
The vectors ${\bf a}$, ${\bf b}$, and ${\bf c}$ are non-coplanar, and
form a non-orthogonal vector base. The vectors ${\bf A}$, ${\bf B}$,
and ${\bf C}$, defined by$$ {\bf A} = \frac{{\bf b}\times {\bf c}}{{\bf a}\cdot{\bf b}\times
{\bf c}}, $$plus cyclic permutations, are said to be reciprocal vectors. Show that
$$ {\bf a} = \frac{{\bf B}\times {\bf C}}{{\bf A}\cdot{\bf B}\times {\bf
C}}, $$plus cyclic permutations.
thanks
Best Answer
Cyclic means a cyclic permutation of the operands: $$ {\bf A} \to {\bf B}, {\bf B} \to {\bf C}, {\bf C} \to {\bf A} \quad a \to b, b \to c, c \to a $$ This gives $$ {\bf A} = \frac{{\bf b}\times {\bf c}}{{\bf a}\cdot({\bf b}\times {\bf c})} \quad {\bf B} = \frac{{\bf c}\times {\bf a}}{{\bf b}\cdot({\bf c}\times {\bf a})} \quad {\bf C} = \frac{{\bf a}\times {\bf b}}{{\bf c}\cdot({\bf a}\times {\bf b})} $$ and $$ {\bf a} = \frac{{\bf B}\times {\bf C}}{{\bf A}\cdot({\bf B}\times {\bf C})} \quad {\bf b} = \frac{{\bf C}\times {\bf A}}{{\bf B}\cdot({\bf C}\times {\bf A})} \quad {\bf c} = \frac{{\bf A}\times {\bf B}}{{\bf C}\cdot({\bf A}\times {\bf B})} $$
From Reciprocal Lattice
Checking the magnitude:
$$ \lVert{\bf A}\rVert = \frac{\lVert {\bf b} \times{\bf c}\rVert}{\lVert{\bf a}\rVert\lVert{\bf b}\times {\bf c}\rVert\cos\angle({\bf a}, {\bf b} \times{\bf c})} = \frac{1}{\lVert{\bf a}\rVert ({\bf e_a} \cdot {\bf e}_{{\bf b} \times{\bf c}})} $$
Solving the question:
Using the "bac-cab" rule ${\bf a}\times({\bf b}\times{\bf c}) = {\bf b}({\bf a}\cdot {\bf c}) - {\bf c}({\bf a}\cdot {\bf b})$ we go for the nominator of ${\bf B}\times{\bf C}$: $$ ({\bf c}\times {\bf a})\times({\bf a}\times{\bf b}) = {\bf a}(({\bf c}\times{\bf a})\cdot{\bf b})-{\bf b}(({\bf c}\times{\bf a})\cdot{\bf a})= {\bf a}(({\bf c}\times{\bf a})\cdot{\bf b}) $$ because ${\bf c}\times{\bf a} \perp {\bf a}$. This gives $$ {\bf B}\times{\bf C} = \frac{{\bf a}(({\bf c}\times{\bf a})\cdot{\bf b})}{({\bf b}\cdot({\bf c}\times {\bf a}))({\bf c}\cdot({\bf a}\times {\bf b}))} = \frac{{\bf a}}{{\bf c}\cdot({\bf a}\times {\bf b})} \\ {\bf A}\cdot({\bf B}\times{\bf C}) = \frac{{\bf b}\times {\bf c}}{{\bf a}\cdot({\bf b}\times {\bf c})} \cdot \frac{{\bf a}}{{\bf c}\cdot({\bf a}\times {\bf b})} = \frac{1}{{\bf c}\cdot({\bf a}\times {\bf b})} $$ Dividing those gives the desired result $$ \frac{{\bf B}\times{\bf C}}{{\bf A}\cdot({\bf B}\times{\bf C})} = \frac{{\bf c}\cdot({\bf a}\times {\bf b})\,{\bf a}}{{\bf c}\cdot({\bf a}\times {\bf b})} = {\bf a} $$