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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 Product of Four Vectors | Mathematics (Maths) Class 12 - JEE 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

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE Writing this result in the form of a determinant,

we have Product of Four Vectors | Mathematics (Maths) Class 12 - JEE


(b) Vector Product of Four Vectors:

Consider next the vector product of Product of Four Vectors | Mathematics (Maths) Class 12 - JEE This is a vector at right angles to   Product of Four Vectors | Mathematics (Maths) Class 12 - JEE  and therefore coplanar with  Product of Four Vectors | Mathematics (Maths) Class 12 - JEE Similarly it is coplanar with   Product of Four Vectors | Mathematics (Maths) Class 12 - JEE It must therefore be parallel to the line of intersection of a plane parallel to Product of Four Vectors | Mathematics (Maths) Class 12 - JEE with another parallel to  Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

To express the product in   Product of Four Vectors | Mathematics (Maths) Class 12 - JEE   in terms of  Product of Four Vectors | Mathematics (Maths) Class 12 - JEE regard it as the vector triple product of Product of Four Vectors | Mathematics (Maths) Class 12 - JEEand  Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE  Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Similarly, regarding it as the vector product of   Product of Four Vectors | Mathematics (Maths) Class 12 - JEE we may write it 

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE  Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Equating these two expressions we have a relation between the four vectors Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE ...(3)

Ex.26 Show that , Product of Four Vectors | Mathematics (Maths) Class 12 - JEE


Sol.

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE


Ex.27 Show that Product of Four Vectors | Mathematics (Maths) Class 12 - JEE


So

l.Product of Four Vectors | Mathematics (Maths) Class 12 - JEE


K. VECTOR EQUATIONS

Ex.28 Solve the equation Product of Four Vectors | Mathematics (Maths) Class 12 - JEE


Sol. From the vector product of each member with a, and obtain Product of Four Vectors | Mathematics (Maths) Class 12 - JEE
Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Ex.29 Solve the simultaneous equations  Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

 

Sol. Multiply the first vectorially by Product of Four Vectors | Mathematics (Maths) Class 12 - JEE
which is of the same form as the equation in the preceding example.

Thus  Product of Four Vectors | Mathematics (Maths) Class 12 - JEE
Substitution of this value in the first equation gives Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Ex.30  Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Sol. Multiply scalarly by  Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

 

 

Ex.31 If 

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE then prove that 

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Sol.

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE ...(1)

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE  Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Solving (2) and Product of Four Vectors | Mathematics (Maths) Class 12 - JEE simultaneously we get the desired result.

 

Ex.32 Solve the vector equation in Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Sol.

Taking dot with a =  Product of Four Vectors | Mathematics (Maths) Class 12 - JEE...(1)

Taking cross with a =Product of Four Vectors | Mathematics (Maths) Class 12 - JEE ...(2)

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Ex.33 Express a vector Product of Four Vectors | Mathematics (Maths) Class 12 - JEEas a linear combination of a vector Product of Four Vectors | Mathematics (Maths) Class 12 - JEE and another perpendicular to A and coplanar with Product of Four Vectors | Mathematics (Maths) Class 12 - JEEand Product of Four Vectors | Mathematics (Maths) Class 12 - JEE.

Sol. 

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE is a vector perpendicular to Product of Four Vectors | Mathematics (Maths) Class 12 - JEE and coplanar with Product of Four Vectors | Mathematics (Maths) Class 12 - JEE and Product of Four Vectors | Mathematics (Maths) Class 12 - JEE.

Hence let, 

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE ...(1)

taking dot with  Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

again taking cross with  Product of Four Vectors | Mathematics (Maths) Class 12 - JEE 

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

Product of Four Vectors | Mathematics (Maths) Class 12 - JEE

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

1. What is the product of four vectors?
The product of four vectors refers to the mathematical operation that combines the four vectors together to obtain a scalar or vector quantity. This operation can be performed using different mathematical techniques, such as the dot product or cross product.
2. How is the dot product used to calculate the product of four vectors?
The dot product is a mathematical operation that takes two vectors as input and produces a scalar as output. To calculate the product of four vectors using the dot product, we can first find the dot product of two pairs of vectors and then take the dot product of the resulting vectors.
3. What is the significance of the product of four vectors in physics?
The product of four vectors is often used in physics to calculate quantities such as work, power, and momentum. These quantities are essential in understanding and describing the behavior of various physical systems, including particles, fluids, and electromagnetic fields.
4. Can the product of four vectors be used in engineering applications?
Yes, the product of four vectors has applications in various engineering fields. For example, in structural engineering, it can be used to calculate forces and moments acting on a structure. In electrical engineering, it is used to determine the power and energy in electrical circuits.
5. Are there any limitations or conditions when calculating the product of four vectors?
Yes, there are certain conditions and limitations when calculating the product of four vectors. For instance, the vectors must be compatible in terms of their dimensions and units. Additionally, certain mathematical properties and rules, such as commutativity and distributivity, may apply depending on the specific operation used for the vector product.
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