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