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.