Product of Four Vectors JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : Product of Four Vectors JEE Notes | EduRev

The document Product of Four Vectors JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 12.
All you need of JEE at this link: 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 Product of Four Vectors JEE Notes | EduRev 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 JEE Notes | EduRev Writing this result in the form of a determinant,

we have Product of Four Vectors JEE Notes | EduRev


(b) Vector Product of Four Vectors:

Consider next the vector product of Product of Four Vectors JEE Notes | EduRev This is a vector at right angles to   Product of Four Vectors JEE Notes | EduRev  and therefore coplanar with  Product of Four Vectors JEE Notes | EduRev Similarly it is coplanar with   Product of Four Vectors JEE Notes | EduRev It must therefore be parallel to the line of intersection of a plane parallel to Product of Four Vectors JEE Notes | EduRev with another parallel to  Product of Four Vectors JEE Notes | EduRev

To express the product in   Product of Four Vectors JEE Notes | EduRev   in terms of  Product of Four Vectors JEE Notes | EduRev regard it as the vector triple product of Product of Four Vectors JEE Notes | EduRevand  Product of Four Vectors JEE Notes | EduRev

Product of Four Vectors JEE Notes | EduRev  Product of Four Vectors JEE Notes | EduRev

Similarly, regarding it as the vector product of   Product of Four Vectors JEE Notes | EduRev we may write it 

Product of Four Vectors JEE Notes | EduRev  Product of Four Vectors JEE Notes | EduRev

Equating these two expressions we have a relation between the four vectors Product of Four Vectors JEE Notes | EduRev

Product of Four Vectors JEE Notes | EduRev ...(3)

Ex.26 Show that , Product of Four Vectors JEE Notes | EduRev


Sol.

Product of Four Vectors JEE Notes | EduRev


Ex.27 Show that Product of Four Vectors JEE Notes | EduRev


So

l.Product of Four Vectors JEE Notes | EduRev


K. VECTOR EQUATIONS

Ex.28 Solve the equation Product of Four Vectors JEE Notes | EduRev


Sol. From the vector product of each member with a, and obtain Product of Four Vectors JEE Notes | EduRev
Product of Four Vectors JEE Notes | EduRev

Ex.29 Solve the simultaneous equations  Product of Four Vectors JEE Notes | EduRev

 

Sol. Multiply the first vectorially by Product of Four Vectors JEE Notes | EduRev
which is of the same form as the equation in the preceding example.

Thus  Product of Four Vectors JEE Notes | EduRev
Substitution of this value in the first equation gives Product of Four Vectors JEE Notes | EduRev

Ex.30  Product of Four Vectors JEE Notes | EduRev

Sol. Multiply scalarly by  Product of Four Vectors JEE Notes | EduRev

Product of Four Vectors JEE Notes | EduRev

Product of Four Vectors JEE Notes | EduRev

Product of Four Vectors JEE Notes | EduRev

Product of Four Vectors JEE Notes | EduRev

 

 

Ex.31 If 

Product of Four Vectors JEE Notes | EduRev then prove that 

Product of Four Vectors JEE Notes | EduRev

Sol.

Product of Four Vectors JEE Notes | EduRev

Product of Four Vectors JEE Notes | EduRev ...(1)

Product of Four Vectors JEE Notes | EduRev

Product of Four Vectors JEE Notes | EduRev  Product of Four Vectors JEE Notes | EduRev

Solving (2) and Product of Four Vectors JEE Notes | EduRev simultaneously we get the desired result.

 

Ex.32 Solve the vector equation in Product of Four Vectors JEE Notes | EduRev

Sol.

Taking dot with a =  Product of Four Vectors JEE Notes | EduRev...(1)

Taking cross with a =Product of Four Vectors JEE Notes | EduRev ...(2)

Product of Four Vectors JEE Notes | EduRev

Product of Four Vectors JEE Notes | EduRev

Ex.33 Express a vector Product of Four Vectors JEE Notes | EduRevas a linear combination of a vector Product of Four Vectors JEE Notes | EduRev and another perpendicular to A and coplanar with Product of Four Vectors JEE Notes | EduRevand Product of Four Vectors JEE Notes | EduRev.

Sol. 

Product of Four Vectors JEE Notes | EduRev is a vector perpendicular to Product of Four Vectors JEE Notes | EduRev and coplanar with Product of Four Vectors JEE Notes | EduRev and Product of Four Vectors JEE Notes | EduRev.

Hence let, 

Product of Four Vectors JEE Notes | EduRev ...(1)

taking dot with  Product of Four Vectors JEE Notes | EduRev

Product of Four Vectors JEE Notes | EduRev

again taking cross with  Product of Four Vectors JEE Notes | EduRev 

Product of Four Vectors JEE Notes | EduRev

Product of Four Vectors JEE Notes | EduRev

Product of Four Vectors JEE Notes | EduRev

Product of Four Vectors JEE Notes | EduRev

Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!

Related Searches

Objective type Questions

,

study material

,

Semester Notes

,

pdf

,

Previous Year Questions with Solutions

,

Exam

,

ppt

,

Summary

,

video lectures

,

Product of Four Vectors JEE Notes | EduRev

,

Important questions

,

shortcuts and tricks

,

Product of Four Vectors JEE Notes | EduRev

,

Product of Four Vectors JEE Notes | EduRev

,

Viva Questions

,

MCQs

,

Sample Paper

,

Extra Questions

,

practice quizzes

,

past year papers

,

Free

,

mock tests for examination

;