The scalars `l` and `m` such that `l vec a+m vec b= vec c` ,where `vec a , vec b` and ` vec c` are given vectors, are equal to

Prove that [[ vec a+ vec b, vec b+ vec c, vec c+ vec a]]=2[ [vec a, vec b, vec c]]dot

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 ba n d vec c are two-noncollinear vectors such that vec a||( vec bxx vec c), then prove that ( vec axx vec b) . ( vec axx vec c) is equal to | vec a|^2( vec bdot vec c)dot

If vec a+ vec b+ vec c = vec 0 then prove that vec axx vec b= vec bxx vec c = vec cxxvec a .

If vec axx vec b= vec bxx vec c!=0,w h e r e vec a , vec b ,a n d vec c are coplanar vectors, then for some scalar k prove that vec a+ vec c=k vec bdot

If vec a,vec b, vec c be three vectors such that vec a+vec b+vec c=0 , |veca|=3 , |vecb|=5 and |vecc|=7 find the angle between the vectors veca and vec b .

If vec a , vec b , and vec c are three vectors such that vec axx vec b= vec c , vec bxx vec c= vec a , vec cxx vec a= vec b , then prove that | vec a|=| vec b|=| vec c| .

If vec a , vec b ,and vec c are non-coplanar unit vectors such that vec axx( vec bxx vec c)=( vec b+ vec c)/(sqrt(2)), vec b and vec c are non-parallel, then prove that the angle between vec a and vec b, is 3pi//4.

[( vec axx vec b)xx( vec bxx vec c)( vec bxx vec c)xx( vec cxx vec a)( vec cxx vec a)xx( vec axx vec b)] is equal to (where vec a , vec ba n d vec c are nonzero non-coplanar vector) a. [ vec a vec b vec c]^2 b. [ vec a vec b vec c]^3 c. [ vec a vec b vec c]^4 d. [ vec a vec b vec c]

Show that the vectors 2 vec a- vec b+3 vec c , vec a+ vec b-2 vec ca n d vec a+ vec b-3 vec c are non-coplanar vectors (where vec a , vec b , vec c are non-coplanar vectors)