Properties of Scalar Triple Products
1 . The scalar triple product is independent of the positions of dot and cross i . e ., ( a * b ) * c = a * ( b * c ).
2 . The scalar triple product of three vectors is unaltered so long as the cyclic order of the vectors remains unchanged .
i . e ., ( a * b ) * c = ( b * c ) * a = ( c * a ) * b or [ a b c ] = [ b c a ] = [ c a b ].
3 . The scalar triple product changes in sign but not in magnitude , when the cyclic order is changed .
i . e ., [ a b c ] = – [ a c b ] etc .
4 . The scalar triple product vanishes , if any two of its vectors are equal . i . e ., [ a a b ] = 0 , [ a b a ] = 0 and [ b a a ] = 0 .
5 . The scalar triple product vanishes , if any two of its vectors are parallel or collinear .
6 . For any scalar x , [ x a b c ] = x [ a b c ]. Also , [ x a yb zc ] = xyz [ a b c ].
7 . For any vectors a , b , c , d , [ a + b c d ] = [ a c d ] + [ b c d ]
8 . [ i j k ] = 1