For any four vectors, prove that `( vec bxx vec c)dot( vec axx vec d)+( vec cxx vec a)dot( vec bxx vec d)+( vec axx vec b)dot( vec cxx vec d)=0.`

If vec a , vec ba n d vec c are three non coplanar vectors, then prove that vec d=( vec adot vec d)/([ vec a vec b vec c])( vec bxx vec c)+( vec bdot vec d)/([ vec a vec b vec c])( vec cxx vec a)+( vec cdot vec d)/([ vec a vec b vec c])( vec axx vec b)

If a ,ba n dc are non-cop0lanar vector, then that prove |( vec adot vec d)( vec bxx vec c)+( vec bdot vec d)( vec cxx vec a)+( vecc dot vec d)( vec axx vec b)| is independent of d ,w h e r ee is a unit vector.

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

For any four vectors,vec a,vec b,vec c and vec d prove that vec d*(vec a xx(vec b xx(vec c xxvec d)))=(vec b*vec d)[vec avec cvec d]

For any three vectors adotb\ a n d\ c write the value of vec axx( vec b+ vec c)+ vec bxx( vec c+ vec a)+ vec cxx( vec a+ vec b)dot

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

Let vec a , vec b , a n d vec c be three non-coplanar vectors and vec d be a non-zero vector, which is perpendicular to ( vec a+ vec b+ vec c)dot Now vec d=( vec axx vec b)sinx+( vec bxx vec c)cosy+2( vec cxx vec a)dotT h e n a.( vec ddot( vec a+ vec b))/([ vec a vec b vec c])=2 b.( vec ddot( vec a+ vec b))/([ vec a vec b vec c])=-2 c. minimum value of x^2+y^2 is pi^2//4 d. minimum value of x^2+y^2 is 5pi^2//4

For any two vectors vec a\ a n d\ vec b , fin d\ ( vec axx vec b). vecbdot

[vec a, vec b + vec c, vec d] = [vec a, vec b, vec d] + [vec a, vec c, vec d]

If vectors vec a , vec b ,a n d vec c are coplanar, show that | vec a vec b vec c vec adot vec a vec adot vec b vec adot vec c vec bdot vec a vec bdot vec b vec bdot vec c|=odot