If `vece_(1),vece_(2),vece_(3)and vecE_(1),vecE_(2),vecE_(3)` are two sets of vectors such that `vece_(i).vecE_(j)=1, if i=j and vece_(i).vecE_(j)=0 andif i nej, " the prove that " [vece_(1)vece_(2)vece_(3)][vecE_(1)vecE_(2)vecE_(3)]=1`

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 A^(3)=O , then prove that (I-A)^(-1) =I+A+A^(2) .

If theta be the angle between two unit vectors vece_1 and vece_2 then prove that |vece_1 - vece_2| = 2 sin (theta/2)

Let vec a=a_1 hat i+a_2 hat j+a_3 hat k , vec b=b_1 hat i+b_2 hat j+b_3 hat k and vec c=c_1 hat i+c_2 hat j+c_3 hat k be three non-zero vectors such that vec c is a unit vector perpendicular to both vec a and vec b . If the angle between a and b is pi/6, then prove that |[a_1,a_2,a_3],[b_1,b_2,b_3],[c_1,c_2,c_3]|^2=1/4(a_1 ^2+a_2 ^2+a_3 ^2)(b_1 ^2+b_2 ^2+b_3 ^2)

A dipole of moment vecp is placed in a uniform field vecE . The force on the dipole is vecF and the torque is vec(tau) . Then

Electric flux enclosed by a surface A is given by - (A) in_0 int vecE.dvecA (B) 1/in_0 intvecE.dvecA (C) in_0mu_0 intvecE.dvecA (D) intvecE.dvecA

Statement 1: vec a=3 vec i+p vec j+3 vec k and vec b=2 vec i+3 vec j+q vec k are parallel vectors if p=9//2a n dq=2. Statement 2: if vec a=a_1 vec i+a_2 vec j+a_3 vec ka n d vec b=b_1 vec i+b_2 vec j+b_3 vec k are parallel, then (a_1)/(b_1)=(a_2)/(b_2)=(a_3)/(b_3)dot Which of the following Statements is/are correct ?

Construct a 3xx4 matrix, whose elements are given by: (i) a_(ij)=(1)/(2)|-3i+j| (ii) a_(ij)=2i-j

Let vec A(t) = f_1(t) hat i + f_2(t) hat j and vec B(t) = g(t)hat i+g_2(t) hat j,t in [0,1],f_1,f_2,g_1, g_2 are continuous functions. If vec A(t) and vec B(t) are non-zero vectors for all t and vec A(0) = 2hat i + 3hat j,vec A(1) = 6hat i + 2hat j, vec B(0) = 3hat i + 2hat i and vec B(1) = 2hat i + 6hat j Then,show that vec A(t) and vec B(t) are parallel for some t .

What will be the electric flux passing through the area vecS=10hati placed in an electric field of intensity vecE=3hati+4hatj+5hatk ?