Triple Product of three vectors

There are two types of triple product of vectors:

(a)   Scalar triple product.

(b)   Vector triple product.

Scalar Triple Product

Definition :

              let u , v, and w  be three vectors.

The scalar triple product of vectors u , v, and w  is defined by u . ( v ×w) or v . ( w ×u)

or w . (u×v).

The scalar triple product u.(v×w) is written as   u. ( v×w) = [u v w ].

Note :

 (1) The value of the scalar triple product depends upon the cycle  order of the vectors , but is independent of the position of the dot and cross. So the dot and cross , may be interchanged without altering the value  i.e;

               u . ( v ×w )= u× ( v .w ) =  [u v w]

  (2) the value of the product changes if the order  is non cyclic i.e;

   u . ( v ×w)= - u.( w×v ) = -[u v w]

Analytical Expression of u .( v ×w):                                

               

         

which is called the determinant formula of scalar triple product of u , v and w in component form.

where u = a₁i + b₁j + c₁k , v = a₂i + b₂j + c₂k and w = a₃i + b₃j + c₃k

THE VOLUME OF THE PARALLELEPIPED

Statement :

The scalar triple product u× ( v .w) represents  the volume of the parallelepiped having u , v and w as its conterminous edges.

Proof: As ,

  u× ( v .w) =  u× v       w  cosӨ

where u ×v    = area of the parallelogram with two adjacent sides u and v.

 and    w  cosӨ  = height of the parallelepiped.

Therefore,   u× ( v .w) =  u × v     w  cosӨ    =( area of parallelepiped)( height)

                                   = volume of parallelepiped.

Similarly by taking the base planeformed by v and w , we have

The volume of parallelepiped  =  (v×  w) . u

And by taking the base plane formed by w and u ,we have

The volume of parallelepiped = (   w ×  u)  .v

Thus ( u  × v  )  .w  =( v  ×w)  .u  =  (w ×  u )  .v   =  V