(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
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.