If ` hat a , hat b ,a n d hat c` are three unit vectors inclined to each other at angle `theta,` then the minimum value of `theta` is `pi/3` b. `pi/4` c. `(2pi)/3` d. `(5pi)/6`

If hat a , hat b ,a n d hat c are unit vectors, then | hat a-hat b|^2+| hat b- hat c|^2+| hat c- hat a|^2 does not exceed

If hat a , hat b ,a n d hat c are three unit vectors, such that hat a+ hat b+ hat c is also a unit vector and theta_1,theta_2 and theta_3 are angles between the vectors hat a , hat b ; hat b , hat ca n d hat c , hat a respectively, then among theta_1,theta_2 and theta_3 . a. all are acute angles b. all are right angles c. at least one is obtuse angle d. none of these

L_1a n dL_2 are two lines whose vector equations are L_1: vec r=lambda((costheta+sqrt(3)) hat i+(sqrt(2)sintheta) hat j+(costheta-sqrt(3)) hat k) L_2: vec r=mu(a hat i+b hat j+c hat k) , where lambdaa n dmu are scalars and alpha is the acute angel between L_1a n dL_2dot If the angel alpha is independent of theta, then the value of alpha is a. pi/6 b. pi/4 c. pi/3 d. pi/2

If 2|sin2alpha|=|tanbeta+cotbeta|,alpha,beta in (pi/2,pi), then the value of alpha+beta is (a) (3pi)/4 (b) pi (c) (3pi)/2 (d) (5pi)/4

Let vec A be a vector parallel to the line of intersection of planes P_1a n dP_2dot Plane P_1 is parallel to vectors 2 hat j+3 hat ka n d4 hat j-3ka n dP_2 is parallel to hat j- hat ka n d3 hat i+3 hat jdot Then the angle betweenvector vec A and a given vector 2 hat i+ hat j-2 hat k is a. pi//2 b. pi//4 c. pi//6 d. 3pi//4

denote the positive solution of the equation 3+3costheta=2sin^2thetadot The value of theta_3+theta_7 is 3 pi (b) 4 pi (c) 5 pi (d) 6 pi

Vector vec a in the plane of vec b=2 hat i+ hat ja n d vec c= hat i- hat j+ hat k is such that it is equally inclined to vec ba n d vec d where vec d= hat j+2 hat kdot The value of vec a is a. ( hat i+ hat j+ hat k)/(sqrt(2)) b. ( hat i- hat j+ hat k)/(sqrt(3)) c. (2 hat i+ hat j)/(sqrt(5)) d. (2 hat i+ hat j)/(sqrt(5))

If 4 hat i+7 hat j+8 hat k ,2 hat i+3 hat j+4hat ja n d2 hat i+5 hat j+7 hat k are the position vectors of the vertices A ,Ba n dC , respectively, of triangle A B C , then the position vecrtor of the point where the bisector of angle A meets B C is a. 2/3(-6 hat i-8 hat j- hat k) b. 2/3(6 hat i+8 hat j+6 hat k) c. 1/3(6 hat i+13 hat j+18 hat k) d. 1/3(5 hat j+12 hat k)

Let vec a* vec b=0,w h e r e vec aa n d vec b are unit vectors and the unit vector vec c is inclined at an angle theta to both vec aa n d vec bdot If vec c=m vec a+n vec b+p( vec axx vec b),(m ,n , p in R), then a. -pi/4lt=thetalt=pi/4 b. pi/4lt=thetalt=(3pi)/4 c. 0lt=thetalt=pi/4 d. 0lt=thetalt=(3pi)/4

If the vectors 2 hat i-3 hat j , hat i+ hat j- hat ka n d3 hat i- hat k form three concurrent edges of a parallelepiped, then find the volume of the parallelepiped.