In March 2025
Answer to Question #110923 in Analytic Geometry for Proloy nag
a×(b×c)+b×(c×a)+c×(a×b)=0
"{\\rm Consider}\\, \\\\\n\\vec{a}=(a_x,a_y,a_z); \\vec{b}=(b_x,b_y,b_z); \\vec{c}=(c_x,c_y,c_z).\\\\\n(\\vec{b}\\times\\vec{c})=\n\\begin{vmatrix}\n \\vec{i} & \\vec{j} & \\vec{k} \\\\\n b_x & b_y & b_z \\\\\nc_x & c_y & c_z\n\\end{vmatrix} \n= \\vec{i} (b_y c_z - b_z c_y )- \\\\\n\\vec{j} ( b_x c_z - b_z c_x ) + \\vec{k} (b_x c_y -b_y c_x);\\\\\n\\vec{a}\\times(\\vec{b}\\times\\vec{c})=\\\\\n\\begin{vmatrix}\n \\vec{i} & \\vec{j} & \\vec{k} \\\\\n a_x & a_y & a_z \\\\\n(b_y c_z - b_z c_y )& ( b_z c_x - b_x c_z ) & (b_x c_y -b_y c_x)\n\\end{vmatrix} =\\\\\n\\vec{i}(a_y (b_x c_y -b_y c_x)-a_z( b_z c_x - b_x c_z ) )-\\\\\n\\vec{j}( a_x(b_x c_y -b_y c_x)-a_z(b_y c_z - b_z c_y ))+\\\\\n\\vec{k}( a_x( b_z c_x - b_x c_z )-a_y(b_y c_z - b_z c_y ));\\\\\n(\\vec{c}\\times\\vec{a})=\n\\begin{vmatrix}\n \\vec{i} & \\vec{j} & \\vec{k} \\\\\n c_x & c_y & c_z \\\\\na_x & a_y & a_z\n\\end{vmatrix} \n= \\vec{i} (c_y a_z - c_z a_y )- \\\\\n\\vec{j} ( c_x a_z - c_z a_x ) + \\vec{k} (c_x a_y -c_y a_x);\\\\\n\\vec{b}\\times(\\vec{c}\\times\\vec{a})=\\\\\n\\begin{vmatrix}\n \\vec{i} & \\vec{j} & \\vec{k} \\\\\n b_x & b_y & b_z \\\\\n(c_y a_z - c_z a_y )& ( c_z a_x - c_x a_z ) & (c_x a_y -c_y a_x)\n\\end{vmatrix} =\\\\\n\\vec{i}(b_y (c_x a_y -c_y a_x)-b_z( c_z a_x - c_x a_z ) )-\\\\\n\\vec{j}( b_x(c_x a_y -c_y a_x)-b_z(c_y a_z - c_z a_y ))+\\\\\n\\vec{k}( b_x( c_z a_x - c_x a_z )-b_y(c_y a_z - c_z a_y ));\\\\\n(\\vec{a}\\times\\vec{b})=\n\\begin{vmatrix}\n \\vec{i} & \\vec{j} & \\vec{k} \\\\\n a_x & a_y & a_z \\\\\nb_x & b_y & b_z\n\\end{vmatrix} \n= \\vec{i} (a_y b_z - a_z b_y )- \\\\\n\\vec{j} ( a_x b_z - a_z b_x ) + \\vec{k} (a_x b_y -a_y b_x);\\\\\n\\vec{c}\\times(\\vec{a}\\times\\vec{b})=\\\\\n\\begin{vmatrix}\n \\vec{i} & \\vec{j} & \\vec{k} \\\\\n c_x & c_y & c_z \\\\\n(a_y b_z - a_z b_y )& ( a_z b_x - a_x b_z ) & (a_x b_y -a_y b_x)\n\\end{vmatrix} =\\\\\n\\vec{i}(c_y (a_x b_y -a_y b_x)-c_z( a_z b_x - a_x b_z ) )-\\\\\n\\vec{j}( c_x(a_x b_y -a_y b_x)-c_z(a_y b_z - a_z b_y ))+\\\\\n\\vec{k}( c_x( a_z b_x - a_x b_z )-c_y(a_y b_z - a_z b_y ));\\\\\n\n\n\n\n\\vec{a}\\times(\\vec{b}\\times\\vec{c})+\\vec{b}\\times(\\vec{a}\\times\\vec{a})+\\vec{c}\\times(\\vec{a}\\times\\vec{b})=\\\\\n(a_y (b_x c_y -b_y c_x)-a_z( b_z c_x - b_x c_z ) )+\\\\\n(b_y (c_x a_y -c_y a_x)-b_z( c_z a_x - c_x a_z ) )+\\\\\n(c_y (a_x b_y -a_y b_x)-c_z( a_z b_x - a_x b_z ) )-\\\\\n(( a_x(b_x c_y -b_y c_x)-a_z(b_y c_z - b_z c_y ))+\\\\\n( b_x(c_x a_y -c_y a_x)-b_z(c_y a_z - c_z a_y ))+\\\\\n( c_x(a_x b_y -a_y b_x)-c_z(a_y b_z - a_z b_y ))\n)+\\\\\n( a_x( b_z c_x - b_x c_z )-a_y(b_y c_z - b_z c_y ))+\\\\\n( b_x( c_z a_x - c_x a_z )-b_y(c_y a_z - c_z a_y ))+\\\\\n( c_x( a_z b_x - a_x b_z )-c_y(a_y b_z - a_z b_y ))=0"
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