Product of Four Vectors | Mathematics (Maths) for JEE Main & Advanced PDF Download

Product of Four Vectors

(a) Scalar Product of Four Vectors: The products already considered are usually sufficient for practical applications. But we occasionally meet with products of four vectors of the following types. Consider the scalar product of  This is a number easily expressible in terms of the scalar products of the individual vectors. For, in virtue of the fact that in a scalar triple product the dot and cross may be interchanged, we may write

 Writing this result in the form of a determinant,

we have

(b) Vector Product of Four Vectors:

Consider next the vector product of  This is a vector at right angles to     and therefore coplanar with   Similarly it is coplanar with    It must therefore be parallel to the line of intersection of a plane parallel to  with another parallel to 

To express the product in      in terms of   regard it as the vector triple product of and 

Similarly, regarding it as the vector product of    we may write it 

Equating these two expressions we have a relation between the four vectors

 ...(3)

Example: Show that ,

Sol.

Example: Show that 

Sol:

Vector Equations

Example: Solve the equation

Sol. From the vector product of each member with a, and obtain

Example: Solve the simultaneous equations  

 Sol. Multiply the first vectorially by
which is of the same form as the equation in the preceding example.

Thus  
Substitution of this value in the first equation gives

Example:  

Sol. Multiply scalarly by  

Example:   then prove that 

Sol. 

 ...(1)

Solving (2) and  simultaneously we get the desired result.

Example: Solve the vector equation in 

Sol. Taking dot with a =  ...(1)

Taking cross with a = ...(2)

Example: Express a vector as a linear combination of a vector  and another perpendicular to A and coplanar with and .

Sol.  is a vector perpendicular to  and coplanar with  and .

Hence let, 

 ...(1)

taking dot with 

again taking cross with