Find 3-dimensional vectors `vec v_1,vec v_2,vec v_3` satisfying `vec v_1* vec v_1=4,vec v_1* vec v_2=-2,vec v_1* vec v_3=6,`
`vec v_2* vec v_2=2 , vec v_2 *vec v_3=-5,vec v_3* vec v_3=29`

Let vec u , vec v and vec w be vector such vec u+ vec v+ vec w= vec0 . If | vec u|=3,| vec v|=4 and | vec w|=5, then find vec u . vec v+ vec v . vec w+ vec w . vec u .

If vec e_1, vec e_2, vec e_3a n d vec E_1, vec E_2, vec E_3 are two sets of vectors such that vec e_idot vec E_j=1,ifi=ja n d vec e_idot vec E_j=0a n difi!=j , then prove that [ vec e_1 vec e_2 vec e_3][ vec E_1 vec E_2 vec E_3]=1.

Find | vec a- vec b|,\ if:| vec a|=2,\ | vec b|=3\ a n d\ vec adot vec b=4

For any two vectors vec ua n d vec v prove that ( vec udot vec v)^2+| vec uxx vec v|^2=| vec u|^2| vec v|^2a n d ( vec1+| vec u|^2)( vec1+| vec v|^2)=(1- vec udot vec v)^2+| vec u+ vec v+( vec uxx vec v)|^2

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

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

If vec a , vec b and vec c are three non-zero, non coplanar vector vec b_1= vec b-( vec bdot vec a)/(| vec a|^2) vec a , vec c_1= vec c-( vec cdot vec a)/(| vec a|^2) vec a+( vec bdot vec c)/(| vec c|^2) vec b_1 , , c_2= vec c-( vec cdot vec a)/(| vec a|^2) vec a-( vec bdot vec c)/(| vec b_1|^2) , b_1, vec c_3= vec c-( vec cdot vec a)/(| vec c|^2) vec a+( vec bdot vec c)/(| vec c|^2) vec b_1 , vec c_4= vec c-( vec cdot vec a)/(| vec c|^2) vec a=( vec bdot vec c)/(| vec b|^2) vec b_1 then the set of orthogonal vectors is ( vec a , vec b_1, vec c_3) b. ( vec a , vec b_1, vec c_2) c. ( vec a , vec b_1, vec c_1) d. ( vec a , vec b_2, vec c_2)

Prove that four points 2 vec a+3 vec b- vec c , vec a-2 vec b+3 vec c ,3 vec a+4 vec b-2 vec c and vec a-6 vec b+6 vec c are coplanar.

If vec e_1, vec e_2, vec e_3a n d vec E_1, vec E_2, vec E_3 are two sets of vectors such that vec e_idot vec E_j=1,ifi=j and vec e_idot vec E_j=0a n difi!=j , then prove that [ (vec e_1, vec e_2 ,vec e_3)][ (vec E_1, vec E_2, vec E_3) ]=1.

If vec u , vec v and vec w are three non-coplanar vectors, then prove that ( vec u+ vec v- vec w) . [ [( vec u- vec v)xx( vec v- vec w)]]= vec u . vec v xx vec w