Vector Triple Product JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : Vector Triple Product JEE Notes | EduRev

The document Vector Triple Product JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 12.
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 (b) Vector Triple Product

Consider next the cross product of  Vector Triple Product JEE Notes | EduRev

This is a vector perpendicular to both a  Vector Triple Product JEE Notes | EduRev is normal to the plane of  Vector Triple Product JEE Notes | EduRev soVector Triple Product JEE Notes | EduRev must lie in this plane. It is therefore expressible in terms of Vector Triple Product JEE Notes | EduRev in the form  Vector Triple Product JEE Notes | EduRev To find the actual expression for   Vector Triple Product JEE Notes | EduRev consider unit vectors  j^ and k^  the first parallel to Vector Triple Product JEE Notes | EduRev and the second perpendicular to it in the plane  Vector Triple Product JEE Notes | EduRev

In terms of  j^ and k^ and the other unit vector î of the right-handed system, the remaining vector  Vector Triple Product JEE Notes | EduRev be written  Vector Triple Product JEE Notes | EduRev Then  Vector Triple Product JEE Notes | EduRev  and the triple product 

Vector Triple Product JEE Notes | EduRev  Vector Triple Product JEE Notes | EduRev

This is the required expression for Vector Triple Product JEE Notes | EduRevin terms of Vector Triple Product JEE Notes | EduRev

Similarly the triple product   Vector Triple Product JEE Notes | EduRev ...(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   Vector Triple Product JEE Notes | EduRev is a vector expressible in terms of   Vector Triple Product JEE Notes | EduRev is one expressible in terms of Vector Triple Product JEE Notes | EduRevThe products in general therefore represent different vectors. If a vector r is resolved into two others in the plane of   Vector Triple Product JEE Notes | EduRev one parallel to and the other perpendicular to it, the former is  Vector Triple Product JEE Notes | EduRev  and therefore the latter  Vector Triple Product JEE Notes | EduRev

Geometrical Interpretation of Vector Triple Product JEE Notes | EduRev 

Consider the expression Vector Triple Product JEE Notes | EduRevwhich itself is a vector, since it is a cross product of two vectors  Vector Triple Product JEE Notes | EduRev Now Vector Triple Product JEE Notes | EduRevis a vector perpendicular to the plane containing  Vector Triple Product JEE Notes | EduRev vector perpendicular to the plane  Vector Triple Product JEE Notes | EduRev therefore Vector Triple Product JEE Notes | EduRev is a vector lies in the plane of Vector Triple Product JEE Notes | EduRevand perpendicular to a . Hence we can express  Vector Triple Product JEE Notes | EduRev in terms of Vector Triple Product JEE Notes | EduRev i.e.  Vector Triple Product JEE Notes | EduRev where x & y are scalars.

Vector Triple Product JEE Notes | EduRev

 

Ex.24 Find a vector Vector Triple Product JEE Notes | EduRev and is orthogonal to the vector  Vector Triple Product JEE Notes | EduRev It is given that the projection of Vector Triple Product JEE Notes | EduRev

Sol.

A vector coplanar with  Vector Triple Product JEE Notes | EduRev is parallel to the triple product,

Vector Triple Product JEE Notes | EduRev

Vector Triple Product JEE Notes | EduRev

Vector Triple Product JEE Notes | EduRev

Ex.25 ABCD is a tetrahedron with A(–5, 22, 5); B(1, 2, 3); C(4, 3, 2); D(–1, 2, –3). Find Vector Triple Product JEE Notes | EduRev What can you say about the values of  Vector Triple Product JEE Notes | EduRev 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.

Vector Triple Product JEE Notes | EduRev

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