Vector Triple Product

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

(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

Sol.

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.

Sol.

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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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.

Mathematics (Maths) Class 12

205 videos|264 docs|139 tests

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