Prove that vectors ` vec u=(a l+a_1l_1) hat i+(a m+a_1m_1) hat j+(a n+a_1n_1) hat k` ` vec v=(b l+b_1l_1) hat i+(b m+b_1m_1) hat j+(b n+b_1n_1) hat k` ` vec w=(b l+b_1l_1) hat i+(b m+b_1m_1) hat j+(b n+b_1n_1) hat k` are coplanar.

Two balls , having linear momenta vec(p)_(1) = p hat(i) and vec(p)_(2) = - p hat(i) , undergo a collision in free space. There is no external force acting on the balls. Let vec(p)_(1) and vec(p)_(2) , be their final momenta. The following option(s) is (are) NOT ALLOWED for any non -zero value of p , a_(1) , a_(2) , b_(1) , b_(2) , c_(1) and c_(2) (i) vec(p)_(1) = a_(1) hat(i) + b_(1) hat(j) + c_(1) hat(k) , vec(p)_(2) = a_(2) hat(i) + b_(2) hat(j) (ii) vec(p)_(1) = c_(1) vec(k) , vec(p)_(2) = c_(2) hat(k) (iii) vec(p)_(1) = a_(1) hat(i) + b_(1) hat(j) + c_(1) hat(k) ,vec(p)_(2) = a_(2) hat(i) + b_(2) hat(j) - c_(1) hat(k) (iv) vec(p)_(1) = a_(1) hat(i) + b_(1) hat(j) , vec(p)_(2) = a_(2) hat(i) + b_(1) hat(j)

vec a = a_1 hat i a_2 hat j a_3 hat k |a| = 1 , &, vec a. vec b = 2 vec b = b_1 hat i b_2 hat j b_3 hat k |b| = 4 vec c = 2 (vec a times vec b) - 3 vec b then angel between vec c and vec b

If 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\ a n d\ vec c=c_1 hat i+c_2 hat j+c_3 hat k , then verify that vec axx( vec b+ vec c)= vec axx vec b+ vec axx vec c

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 ; vec c= c_1hat 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 & vec b. If the angle between vec a and vec b is pi/6 , then |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|^2=

Let V be the volume of the parallelepiped formed by the vectors vec a=a_(i)hat i+a_(2)hat j+a_(3)hat k and vec b=b_(1)hat i+bhat j+b_(3)hat k and vec c=c_(1)hat i+c_(2)hat j+c_(3)hat k. If a_(r),b_(r) and cr, where r=1,2,3, are non-negative real numbers and sum_(r=1)^(3)(a_(r)+b_(r)+c_(r))=3L show that V<=L^(3)

The equation of the line passing through the points a_1 hat i+a_2 hat j+a_3 hat k and b_1 hat i+b_2 hat j+b_3 hat k is -> r=(a_1 hat i+a_2 hat j+a_3 hat k)+lambda(b_1 hat i+b_2 hat j+b_3 hat k) -> r=(a_1 hat i+a_2 hat j+a_3 hat k)-t(b_1 hat i+b_2 hat j+b_3 hat k) -> r=a_1(1-t) hat i+_2(1-t) hat j+a_3(1-t) hat k+t(b_1 hat i+b_2 hat j+b_3 hat k)dot None of these

vec a = (hat i + hat j + hat k), vec a * vec b = 1 and vec a xxvec b = hat j-hat k. Then vec b is

Let overset(to)(a) =a_(1) hat(i) + a_(2) hat(j) + a_(3) hat(k) , overset(to)(a) = b_(1) hat(i) +b_(2) hat(j) +b_(3) hat(k) " and " overset(to)(a) = c_(1) hat(i) +c_(2) hat(j) + c_(3) hat(k) be three non- zero vectors such that overset(to)(c ) is a unit vectors perpendicular to both the vectors overset(to)(c ) and overset(to)(b) . If the angle between overset(to)(a) " and " overset(to)(n) is (pi)/(6) then |{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(c_(1),,c_(2),,c_(3)):}| is equal to

If vec adot hat i=vec ahat i+hat j=vec ahat i+dot hat j+hat k=1, then vec a=vec 0 b.hat i c.hat j d.hat i+hat j+hat k

If vec a= hat i+ hat j+ hat k , vec adot vec b=2 and vec ax vec b=2 hat i+ hat j-3 hat k , then vec a+ vec b=5 hat i-4 hat j+2 hat k (b) vec a+ vec b=3 hat i+2 vec k vec b=2 hat i- hat j+ hat k (d) vec b= hat i-2 hat j-3 hat k