Vector Triple Product - Mathematics (Maths) Class 12 - JEE

 (b) Vector Triple Product

Consider next the cross product of 

This is a vector perpendicular to both a   is normal to the plane of   so must lie in this plane. It is therefore expressible in terms of  in the form   To find the actual expression for    consider unit vectors  j^{^} and k^{^}  the first parallel to  and the second perpendicular to it in the plane 

In terms of  j^{^} and k^{^} and the other unit vector î of the right-handed system, the remaining vector   be written   Then    and the triple product 

This is the required expression for in terms of 

Similarly the triple product    ...(2)

It will be noticed that the expansions (1) and (2) are both written down by the same rule. Each scalar product involves the factor outside the bracket; and the first is the scalar product of the extremes.
In a vector triple product the position of the brackets cannot be changed without altering the value of the product. For    is a vector expressible in terms of    is one expressible in terms of The products in general therefore represent different vectors. If a vector r is resolved into two others in the plane of    one parallel to and the other perpendicular to it, the former is    and therefore the latter 

Geometrical Interpretation of  

Consider the expression which itself is a vector, since it is a cross product of two vectors   Now is a vector perpendicular to the plane containing   vector perpendicular to the plane   therefore  is a vector lies in the plane of and perpendicular to a . Hence we can express   in terms of  i.e.   where x & y are scalars.

Ex.24 Find a vector  and is orthogonal to the vector   It is given that the projection of 

Sol.

A vector coplanar with   is parallel to the triple product,

Ex.25 ABCD is a tetrahedron with A(–5, 22, 5); B(1, 2, 3); C(4, 3, 2); D(–1, 2, –3). Find  What can you say about the values of   Calculate the volume of the tetrahedron ABCD and the vector area of the triangle AEF where the quadrilateral ABDE and quadrilateral ABCF are parallelograms.

Sol.