Home
Class 12
MATHS
Prove that vec R+([ vec Rdot( vecbetaxx...

Prove that ` vec R+([ vec Rdot( vecbetaxx( vecbetaxx vecalpha))] vecalpha)/(| vecalphaxx vecbeta|^2) +([ vec Rdot( vecalphaxx( vecalphaxx vecbeta))] vecbeta)/(| vecalphaxx vecbeta|^2) =([ vec R vecalpha vecbeta]( vecalphaxx vecbeta))/(| vecalphaxx vecbeta|^2)`

Text Solution

Verified by Experts

`vecalpha , vecbeta and vecalpha xx vecbeta` are three non-coplanar vectors. Any vector `vecR` can be respresented as a linear combination of these vectors. Thus ,
`vecR=k_(1)vecalpha+k_(2)vecbeta+k_(3)(vecalphaxxvecbeta)`
Take dot product of (i) with `(vecalpha xx vec beta)` . we have
`vecR.(vecalphaxxvecbeta)=k_(3)(vecalpha xxvecbeta)=k_(3)|vecalphaxxvecbeta|^(2)`
`k_(3)=(vecR.(vecalphaxxvecbeta))/(|vecalphaxxvecbeta|^(2))=([vecRvecalphavecbeta])/(|vecalphaxxvecbeta|^(2))`
Take dot product of (i) with `vecalphaxx(vecalphaxxvecbeta)` we have
`vecR.(vecalphaxx(vecalphaxxvecbeta))=k_(2)(vecalphaxx(vecalphaxxvecbeta)).vecbeta`
`= k_(2)[(vecalpha.vecbeta)vecalpha-(vecalpha.vecalpha)vecbeta].vecbeta=k_(2)[(vecalpha.vecbeta)^(2)-|vecalpha|^(2)|vecbeta|^(2)]`
`=-k_(2)|vecalphaxxvecbeta|^(2)`
`k_(2)=(-[vecR.(vecalphaxx(vecalphaxxvecbeta))])/(|vecalphaxxvecbeta|^(2)) " simiarly "k_(1)=-([vecR.(vecbetaxx(vecbetaxxvecalpha))])/(|vecalphaxx vecbeta|^(2))`
`Rightarrow vecR=(-[vecR.[vecbetaxx(vecbetaxxvecalpha))]vecalpha)/(|vecalphaxxvecbeta|^(2))-([vecR.(vecalphaxx(vecalphaxxvecbeta))]vecbeta)/(|vecalphaxxvecbeta|^(2))+(([vecR.(vecalphaxxvecbeta))](vecalphaxxvecbeta))/(|vecalpha xx vecbeta|^(2))`
`Rightarrow vecR=(-[vecR.[vecbetaxx(vecbetaxxvecalpha))]vecalpha)/(|vecalphaxxvecbeta|^(2))-([vecR.(vecalphaxx(vecalphaxxvecbeta))]vecbeta)/(|vecalphaxxvecbeta|^(2))+(([vecR.(vecalphaxxvecbeta))](vecalphaxxvecbeta))/(|vecalpha xx vecbeta|^(2))`
Promotional Banner

Topper's Solved these Questions

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE PUBLICATION|Exercise Exercise 2.1|18 Videos

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE PUBLICATION|Exercise Exercise 2.2|15 Videos

  • DETERMINANTS

    CENGAGE PUBLICATION|Exercise All Questions|262 Videos

  • DIFFERENTIAL EQUATIONS

    CENGAGE PUBLICATION|Exercise All Questions|578 Videos

Similar Questions

Explore conceptually related problems

if vecalpha||( vecbetaxx vecgamma) , then ( vecalphaxxbeta).(vecalphaxx vecgamma) equals to a. | vecalpha|^2( vecbeta.vecgamma) b. | vecbeta|^2( vecgamma. vecalpha) c. | vecgamma|^2( vecalpha. vecbeta) d. | vecalpha|| vecbeta|| vecgamma|

If a(vecalpha xx vecbeta) + b(vecbeta xx vecgamma) +c(vecgamma xx vecalpha)=veco , where a, b, c are non-zero scalars, then the vectors vecalpha, vecbeta, vecgamma are

If vecalpha+ vecbeta+ vecgamma=a vecdeltaa n d vecbeta+ vecgamma+ vecdelta=b vecalpha, vecalphaa n d vecdelta are non-colliner, then vecalpha+ vecbeta+ vecgamma+ vecdelta equals a. a vecalpha b. b vecdelta c. 0 d. (a+b) vecgamma

Distance of the point P( vec p) from the line vec r= vec a+lambda vec b is a. |( vec a- vec p)+((( vec p- vec a)dot vec b) vec b)/(| vec b|^2)| b. |( vec b- vec p)+((( vec p- vec a)dot vec b) vec b)/(| vec b|^2)| c. |( vec a- vec p)+((( vec p- vec b)dot vec b) vec b)/(| vec b|^2)| d. none of these

If vectors vecalpha,vecbeta, vecgamma are such that vecalpha+vecbeta+vec gamma=vec0 and |veca|=2, |vecbeta|=3 and |vecgamma|=4 , then the value of 2(vecalpha*vecbeta+vecbeta*vecgamma+vecgamma*vecalpha) is -

Let vec a , vec ba n d vec c be three non-coplanar vectors and vec p , vec qa n d vec r the vectors defined by the relation vec p=( vec bxx vec c)/([ vec a vec b vec c]), vec q=( vec cxx vec a)/([ vec a vec b vec c])a n d vec r=( vec axx vec b)/([ vec a vec b vec c])dot Then the value of the expression ( vec a+ vec b)dot vec p+( vec b+ vec c)dot vec q+( vec c+ vec a)dot vec r is a. 0 b. 1 c. 2 d. 3

Let vec a , vec ba n d vec c be three non-coplanar vecrors and vec r be any arbitrary vector. Then ( vec axx vec b)xx( vec rxx vec c)+( vec bxx vec c)xx( vec rxx vec a)+( vec cxx vec a)xx( vec rxx vec b) is always equal to [ vec a vec b vec c] vec r b. 2[ vec a vec b vec c] vec r c. 3[ vec a vec b vec c] vec r d. none of these

Vectors vec Aa n d vec B satisfying the vector equation vec A+ vec B= vec a , vec Axx vec B= vec ba n d vec A*vec a=1,w h e r e vec aa n d vec b are given vectors, are a. vec A=(( vec axx vec b)- vec a)/(a^2) b. vec B=(( vec bxx vec a)+ vec a(a^2-1))/(a^2) c. vec A=(( vec axx vec b)+ vec a)/(a^2) d. vec B=(( vec bxx vec a)- vec a(a^2-1))/(a^2)

If vec aa n d vec b are two vectors and angle between them is theta, then | vec axx vec b|^2+( vec adot vec b)^2=| vec a|^2| vec b|^2 | vec axx vec b|=( vec adot vec b),iftheta=pi//4 vec axx vec b=( vec adot vec b) hat n ,(w h e r e hat n is unit vector, ) if theta=pi//4 ( vec axx vec b)dot( vec a+ vec b)=0