11 . Three non-zero vectors a , b and c are coplanar , if and only if [ a b c ] = 0 .
12 . Four points A , B , C , D with position vectors a , b , c , d respectively are coplanar , if and only if [ AB AC AD ] = 0 .
i . e ., if and only if [ b — a c — a d — a ] = 0 .
13 . Volume of parallelepiped with three coterminous edges a , b , c is | [ a b c ] |.
14 . Volume of prism on a triangular base with three coterminous edges a , b , c is 1 / 2 | [ a b c ] |.
15 . Volume of a tetrahedron with three coterminous edges a , b , c is 1 / 6 | [ a b c ] |.
16 . If a , b , c and d are position vectors of vertices of a tetrahedron , then Volume = 1 / 6 [ b — a c — a d — a ].
Vector Triple Product
If a , b , c be any three vectors , then ( a * b ) * c and a * ( b * c ) are known as vector triple product .
∴ a * ( b * c )= ( a * c ) b — ( a * b ) c and ( a * b ) * c = ( a * c ) b — ( b * c ) a
Important Properties
( i ) The vector r = a * ( b * c ) is perpendicular to a and lies in the plane b and c .
( ii ) a * ( b * c ) ≠ ( a * b ) * c , the cross product of vectors is not associative .
( iii ) a * ( b * c )= ( a * b ) * c , if and only if and only if ( a * c ) b — ( a * b ) c = ( a * c ) b — ( b * c ) a , if and only if c = ( b * c ) / ( a * b ) * a
Or if and only if vectors a and c are collinear .