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,
(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
Ex.26 Show that ,
Ex.27 Show that
K. VECTOR EQUATIONS
Ex.28 Solve the equation
Sol. From the vector product of each member with a, and obtain
Ex.29 Solve the simultaneous equations
Sol. Multiply the first vectorially by
which is of the same form as the equation in the preceding example.
Substitution of this value in the first equation gives
Sol. Multiply scalarly by
then prove that
Solving (2) and simultaneously we get the desired result.
Ex.32 Solve the vector equation in
Taking dot with a = ...(1)
Taking cross with a = ...(2)
Ex.33 Express a vector as a linear combination of a vector and another perpendicular to A and coplanar with and .
is a vector perpendicular to and coplanar with and .
taking dot with
again taking cross with