Home
Class 11
MATHS
for any two vectors vecu and vecv , prov...

for any two vectors `vecu and vecv` , prove that
`a. (vecu.vecv)^(2)+ |vecuxx vecv|^(2) = |vecu|^(2) |vecv|^(2)` and
b.` (vec1 + |vecu|^(2)) (vec1 + |vecv|^(2)| = (1=vecu..vecv)^(2) + |vecu +vecv + (vecu xx vecv)|^(2)`

Text Solution

Verified by Experts

we have `vecu.vecv = |vecu||vecv| cos theta`
`and vecu xx vecv =|vecu||vecv|sin theta hatn`
(where `theta` is the angle between `vecu and hatn` is unit vector perpendicular to both `vecu and vecv)`
`Rightarrow (vecu.vecv)^(2) +|vecu xx vecv|^(2)`
`=|vecu|^(2)|vecv|^(2) (cos^(2)theta+sin^(2)theta)=|vecu|^(2)|vecv|^(2)`
`(1-vecu.vecv)^(2)+|vecu+vecv+(vecuxxvecv)|^(2)`
`=1-2vecu.vecv+(vecu xx vecv)^(2)+|vecu|^(2)`
`+|vecv|^(2)+|vecuxxvecv|^(2)+2vecu.vecv`
`(vecu.(vecuxxvecv)=vecv.(vecuxxvecv)=0)`
`1+ |vecu|^(2)+|vecV|^(2)+(vecu.vecv)^(2)+|vecuxxvecv|^(2)`
`1+|vecu|^(2)+|vecv|^(2) + |vecu|^(2) |vecv|^(2)`
`(vec1+|vecu|^(2))(vec1+|vecv|^(2))`
Promotional Banner

Topper's Solved these Questions

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE|Exercise Exercise 2.1|18 Videos

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE|Exercise Exercise 2.2|15 Videos

  • CONIC SECTIONS

    CENGAGE|Exercise Solved Examples And Exercises|1255 Videos

  • LIMITS AND DERIVATIVES

    CENGAGE|Exercise Solved Examples And Exercises|288 Videos

Similar Questions

Explore conceptually related problems

For any two vectors vec ua n d vec v prove that ( vec u . vec v)^2+| vec uxx vec v|^2=| vec u|^2| vec v|^2

For any two vectors vecu and vecv prove that (1+|vecu|^2(1+|vecv|^20=(1-vecu.vecc)^2+|vecu+vecv+vecuxxvec|^2

For any two vectors vec a and vec b , prove that | vec a xx vec b|^(2) = |vec a|^(2)|vec b|^(2) - (vec a . vecb)^(2) = [[veca.veca veca .vecb], [veca.vecb vec b.vecb]]

If vec a and vec b are orthogonal vectors, prove that (vec a + vec b)^(2) = (vec a - vec b)^(2) .

If vec a_|_ vec b , then vector vec v in terms of vec aa n d vec b satisfying the equation s vec v . vec a=0a n d vec v . vec b=1a n d[ vec v vec a vec b]=1 is a. vec b/(| vec b|^2)+( vec axx vec b)/(| vec axx vec b|^2) b. vec b/(| vec b|^2)+( vec axx vec b)/(| vec axx vec b|^2) c. vec b/(| vec b|^2)+( vec axx vec b)/(| vec axx vec b|^2) d. none of these

Show that (vec a - vec b) xx (vec a + vec b) = 2 (vec a xx vec b)

If vec a = hat i - hat j and vec b = hat j + hat k then |vec a xx vec b|^(2) + |vec a. vecb|^(2) is equal to

The value of |vec(a)+vec(b)|^(2)+|vec(a)-vec(b)|^(2) is

Prove that (vec a + vec b).(vec a + vec b) = | vec a|^(2) + |vec b|^(2) , if and only if vec a, vec b are perpendicualr, given a ne vec 0, b ne vec 0 .

If vec a , vec ba n d vec c are three mutually perpendicular vectors, then the vector which is equally inclined to these vectors is a. vec a+ vec b+ vec c b. vec a/(| vec a|)+ vec b/(| vec b|)+ vec c/(| vec c|) c. vec a/(| vec a|^2)+ vec b/(| vec b|^2)+ vec c/(| vec c|^2) d. | vec a| vec a-| vec b| vec b+| vec c| vec c