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.`

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)

Statement 1: vec a , vec b ,a n d vec c are three mutually perpendicular unit vectors and vec d is a vector such that vec a , vec b , vec ca n d vec d are non-coplanar. If [ vec d vec b vec c]=[ vec d vec a vec b]=[ vec d vec c vec a]=1,t h e n vec d= vec a+ vec b+ vec c Statement 2: [ vec d vec b vec c]=[ vec d vec a vec b]=[ vec d vec c vec a] =>vec d is equally inclined to veca,vecb,vecc.

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

prove that | vec axx vec b|=( vec adot vec b)t a ntheta, w h e r e theta is the angle between vec a a n d vec bdot

If vec x+ vec cxx vec y= vec aa n d vec y+ vec cxx vec x= vec b ,w h e r e vec c is a nonzero vector, then which of the following is not correct? vec x=( vec bxx vec c+ vec a+( vec cdot vec a) vec c)/(1+ vec cdot vec c) b. vec x=( vec cxx vec b+ vec b+( vec cdot vec a) vec c)/(1+ vec cdot vec c) c. vec y=( vec axx vec c+ vec b+( vec cdot vec b) vec c)/(1+ vec cdot vec c) d. none of these

If hat(e_(1)) and hat(e_(2)) are two unit vectors such that vec(e_(1)) - vec(e_(2)) is also a unit vector , then find the angle theta between hat(e_(1)) and hat(e_(2)) .

A vector vec d is equally inclined to three vectors vec a= hat i+ hat j+ hat k , vec b=2 hat i+ hat ja n d vec c=3 hat j-2 hat kdot Let vec x , vec y ,a n d vec z be thre vectors in the plane of vec a , vec b ; vec b , vec c ; vec c , vec a , respectively. Then vec xdot vec d=-1 b. vec ydot vec d=1 c. vec zdot vec d=0 d. vec rdot vec d=0,w h e r e vec r=lambda vec x+mu vec y+delta vec z

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

Given that vec a , vec b , vec p , vec q are four vectors such that vec a+ vec b=mu vec p , vec b*vec q=0a n d|vec b|^2=1,w h e r emu is a scalar. Then |( vec adot vec q) vec p-( vec pdot vec q) vec a| is equal to a.2| vec pdot vec q| b. (1//2)| vec pdot vec q| c. | vec pxx vec q| d. | vec pdot vec q|

If the vectors vec a , vec b ,a n d vec c form the sides B C ,C Aa n dA B , respectively, of triangle A B C ,t h e n vec adot vec b+ vec bdot vec c+ vec cdot vec a=0 b. vec axx vec b= vec bxx vec c= vec cxx vec a c. vec adot vec b= vec bdot vec c= vec cdot vec a d. vec axx vec b+ vec bxx vec c+ vec cxx vec a=0