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

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  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

Writing this result in the form of a determinant,

we have

(b) Vector Product of Four Vectors:

Consider next the vector product of  This is a vector at right angles to     and therefore coplanar with   Similarly it is coplanar with    It must therefore be parallel to the line of intersection of a plane parallel to  with another parallel to

To express the product in      in terms of   regard it as the vector triple product of and

Similarly, regarding it as the vector product of    we may write it

Equating these two expressions we have a relation between the four vectors

...(3)

Ex.26 Show that ,

Sol.

Ex.27 Show that

So

l.

K. VECTOR EQUATIONS

Ex.28 Solve the equation

Sol. From the vector product of each member with a, and obtain

Ex.29 Solve the simultaneous equations

Sol. Multiply the first vectorially by
which is of the same form as the equation in the preceding example.

Thus
Substitution of this value in the first equation gives

Ex.30

Sol. Multiply scalarly by

Ex.31 If

then prove that

Sol.

...(1)

Solving (2) and  simultaneously we get the desired result.

Ex.32 Solve the vector equation in

Sol.

Taking dot with a =  ...(1)

Taking cross with a = ...(2)

Ex.33 Express a vector as a linear combination of a vector  and another perpendicular to A and coplanar with and .

Sol.

is a vector perpendicular to  and coplanar with  and .

Hence let,

...(1)

taking dot with

again taking cross with

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## 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.

## Mathematics (Maths) Class 12

206 videos|263 docs|139 tests

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