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Vector Triple Product | Mathematics (Maths) Class 12 - JEE PDF Download

 (b) Vector Triple Product

Consider next the cross product of  Vector Triple Product | Mathematics (Maths) Class 12 - JEE

This is a vector perpendicular to both a  Vector Triple Product | Mathematics (Maths) Class 12 - JEE is normal to the plane of  Vector Triple Product | Mathematics (Maths) Class 12 - JEE soVector Triple Product | Mathematics (Maths) Class 12 - JEE must lie in this plane. It is therefore expressible in terms of Vector Triple Product | Mathematics (Maths) Class 12 - JEE in the form  Vector Triple Product | Mathematics (Maths) Class 12 - JEE To find the actual expression for   Vector Triple Product | Mathematics (Maths) Class 12 - JEE consider unit vectors  j^ and k^  the first parallel to Vector Triple Product | Mathematics (Maths) Class 12 - JEE and the second perpendicular to it in the plane  Vector Triple Product | Mathematics (Maths) Class 12 - JEE

In terms of  j^ and k^ and the other unit vector î of the right-handed system, the remaining vector  Vector Triple Product | Mathematics (Maths) Class 12 - JEE be written  Vector Triple Product | Mathematics (Maths) Class 12 - JEE Then  Vector Triple Product | Mathematics (Maths) Class 12 - JEE  and the triple product 

Vector Triple Product | Mathematics (Maths) Class 12 - JEE  Vector Triple Product | Mathematics (Maths) Class 12 - JEE

This is the required expression for Vector Triple Product | Mathematics (Maths) Class 12 - JEEin terms of Vector Triple Product | Mathematics (Maths) Class 12 - JEE

Similarly the triple product   Vector Triple Product | Mathematics (Maths) Class 12 - JEE ...(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 | Mathematics (Maths) Class 12 - JEE is a vector expressible in terms of   Vector Triple Product | Mathematics (Maths) Class 12 - JEE is one expressible in terms of Vector Triple Product | Mathematics (Maths) Class 12 - JEEThe products in general therefore represent different vectors. If a vector r is resolved into two others in the plane of   Vector Triple Product | Mathematics (Maths) Class 12 - JEE one parallel to and the other perpendicular to it, the former is  Vector Triple Product | Mathematics (Maths) Class 12 - JEE  and therefore the latter  Vector Triple Product | Mathematics (Maths) Class 12 - JEE

Geometrical Interpretation of Vector Triple Product | Mathematics (Maths) Class 12 - JEE 

Consider the expression Vector Triple Product | Mathematics (Maths) Class 12 - JEEwhich itself is a vector, since it is a cross product of two vectors  Vector Triple Product | Mathematics (Maths) Class 12 - JEE Now Vector Triple Product | Mathematics (Maths) Class 12 - JEEis a vector perpendicular to the plane containing  Vector Triple Product | Mathematics (Maths) Class 12 - JEE vector perpendicular to the plane  Vector Triple Product | Mathematics (Maths) Class 12 - JEE therefore Vector Triple Product | Mathematics (Maths) Class 12 - JEE is a vector lies in the plane of Vector Triple Product | Mathematics (Maths) Class 12 - JEEand perpendicular to a . Hence we can express  Vector Triple Product | Mathematics (Maths) Class 12 - JEE in terms of Vector Triple Product | Mathematics (Maths) Class 12 - JEE i.e.  Vector Triple Product | Mathematics (Maths) Class 12 - JEE where x & y are scalars.

Vector Triple Product | Mathematics (Maths) Class 12 - JEE

 

Ex.24 Find a vector Vector Triple Product | Mathematics (Maths) Class 12 - JEE and is orthogonal to the vector  Vector Triple Product | Mathematics (Maths) Class 12 - JEE It is given that the projection of Vector Triple Product | Mathematics (Maths) Class 12 - JEE

Sol.

A vector coplanar with  Vector Triple Product | Mathematics (Maths) Class 12 - JEE is parallel to the triple product,

Vector Triple Product | Mathematics (Maths) Class 12 - JEE

Vector Triple Product | Mathematics (Maths) Class 12 - JEE

Vector Triple Product | Mathematics (Maths) Class 12 - JEE

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 | Mathematics (Maths) Class 12 - JEE What can you say about the values of  Vector Triple Product | Mathematics (Maths) Class 12 - JEE 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 | Mathematics (Maths) Class 12 - JEE

The document Vector Triple Product | Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on Vector Triple Product - Mathematics (Maths) Class 12 - JEE

1. What is a vector triple product?
Ans. A vector triple product refers to the mathematical operation that combines three vectors to result in a new vector. It is calculated as the cross product of two vectors, which is then crossed with a third vector.
2. How is the vector triple product calculated?
Ans. To calculate the vector triple product, you first take the cross product of two vectors, let's say vectors A and B. Then, you take the resulting vector and cross it with a third vector, let's say vector C. The formula for the vector triple product is (A x B) x C.
3. What is the geometric interpretation of the vector triple product?
Ans. Geometrically, the vector triple product represents the volume of a parallelepiped formed by the three vectors. The magnitude of the vector triple product gives the volume of the parallelepiped, and its direction is perpendicular to the plane containing the three vectors.
4. What are some applications of the vector triple product?
Ans. The vector triple product finds applications in various fields, including physics, engineering, and computer graphics. It is used to calculate moments, torques, angular momentum, and magnetic fields. Additionally, it is utilized in the creation of 3D computer models and animations.
5. Can the vector triple product be commutative?
Ans. No, the vector triple product is not commutative. The order in which the vectors are crossed affects the resulting vector. Changing the order of the vectors in the vector triple product will result in a different direction and magnitude for the resultant vector.
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