Answer to Question #110923 in Analytic Geometry for Proloy nag

Question

show that for any three vectors a,b,c
a×(b×c)+b×(c×a)+c×(a×b)=0

Answer 1

${\rm Consider}\, \\ \vec{a}=(a_x,a_y,a_z); \vec{b}=(b_x,b_y,b_z); \vec{c}=(c_x,c_y,c_z).\\ (\vec{b}\times\vec{c})= \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{vmatrix} = \vec{i} (b_y c_z - b_z c_y )- \\ \vec{j} ( b_x c_z - b_z c_x ) + \vec{k} (b_x c_y -b_y c_x);\\ \vec{a}\times(\vec{b}\times\vec{c})=\\ \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_x & a_y & a_z \\ (b_y c_z - b_z c_y )& ( b_z c_x - b_x c_z ) & (b_x c_y -b_y c_x) \end{vmatrix} =\\ \vec{i}(a_y (b_x c_y -b_y c_x)-a_z( b_z c_x - b_x c_z ) )-\\ \vec{j}( a_x(b_x c_y -b_y c_x)-a_z(b_y c_z - b_z c_y ))+\\ \vec{k}( a_x( b_z c_x - b_x c_z )-a_y(b_y c_z - b_z c_y ));\\ (\vec{c}\times\vec{a})= \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ c_x & c_y & c_z \\ a_x & a_y & a_z \end{vmatrix} = \vec{i} (c_y a_z - c_z a_y )- \\ \vec{j} ( c_x a_z - c_z a_x ) + \vec{k} (c_x a_y -c_y a_x);\\ \vec{b}\times(\vec{c}\times\vec{a})=\\ \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ b_x & b_y & b_z \\ (c_y a_z - c_z a_y )& ( c_z a_x - c_x a_z ) & (c_x a_y -c_y a_x) \end{vmatrix} =\\ \vec{i}(b_y (c_x a_y -c_y a_x)-b_z( c_z a_x - c_x a_z ) )-\\ \vec{j}( b_x(c_x a_y -c_y a_x)-b_z(c_y a_z - c_z a_y ))+\\ \vec{k}( b_x( c_z a_x - c_x a_z )-b_y(c_y a_z - c_z a_y ));\\ (\vec{a}\times\vec{b})= \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix} = \vec{i} (a_y b_z - a_z b_y )- \\ \vec{j} ( a_x b_z - a_z b_x ) + \vec{k} (a_x b_y -a_y b_x);\\ \vec{c}\times(\vec{a}\times\vec{b})=\\ \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ c_x & c_y & c_z \\ (a_y b_z - a_z b_y )& ( a_z b_x - a_x b_z ) & (a_x b_y -a_y b_x) \end{vmatrix} =\\ \vec{i}(c_y (a_x b_y -a_y b_x)-c_z( a_z b_x - a_x b_z ) )-\\ \vec{j}( c_x(a_x b_y -a_y b_x)-c_z(a_y b_z - a_z b_y ))+\\ \vec{k}( c_x( a_z b_x - a_x b_z )-c_y(a_y b_z - a_z b_y ));\\ \vec{a}\times(\vec{b}\times\vec{c})+\vec{b}\times(\vec{a}\times\vec{a})+\vec{c}\times(\vec{a}\times\vec{b})=\\ (a_y (b_x c_y -b_y c_x)-a_z( b_z c_x - b_x c_z ) )+\\ (b_y (c_x a_y -c_y a_x)-b_z( c_z a_x - c_x a_z ) )+\\ (c_y (a_x b_y -a_y b_x)-c_z( a_z b_x - a_x b_z ) )-\\ (( a_x(b_x c_y -b_y c_x)-a_z(b_y c_z - b_z c_y ))+\\ ( b_x(c_x a_y -c_y a_x)-b_z(c_y a_z - c_z a_y ))+\\ ( c_x(a_x b_y -a_y b_x)-c_z(a_y b_z - a_z b_y )) )+\\ ( a_x( b_z c_x - b_x c_z )-a_y(b_y c_z - b_z c_y ))+\\ ( b_x( c_z a_x - c_x a_z )-b_y(c_y a_z - c_z a_y ))+\\ ( c_x( a_z b_x - a_x b_z )-c_y(a_y b_z - a_z b_y ))=0$