If `P` is any arbitrary point on the circumcirlce of the equllateral trangle of side length `l` units, then `| vec P A|^2+| vec P B|^2+| vec P C|^2` is always equal to `2l^2` b. `2sqrt(3)l^2` c. `l^2` d. `3l^2`

If vec ra n d vec s are non-zero constant vectors and the scalar b is chosen such that | vec r+b vec s| is minimum, then the value of |b vec s|^2+| vec r+b vec s|^2 is equal to a. 2| vec r|^2 b. | vec r|^2//2 c. 3| vec r""|^2 d. |r|^2

vec a , vec b , vec c are three coplanar unit vectors such that vec a+ vec b+ vec c=0. If three vectors vec p , vec q ,a n d vec r are parallel to vec a , vec b ,a n d vec c , respectively, and have integral but different magnitudes, then among the following options, | vec p+ vec q+ vec r| can take a value equal to a. 1 b. 0 c. sqrt(3) d. 2

vec a , vec b ,a n d vec c are three vectors of equal magnitude. The angel between each pair of vectors is pi//3 such that | vec a+ vec b+ vec c|=6. Then | vec a| is equal to 2 b. -1 c. 1 d. sqrt(6)//3

If C_1: x^2+y^2=(3+2sqrt(2))^2 is a circle and P A and P B are a pair of tangents on C_1, where P is any point on the director circle of C_1, then the radius of the smallest circle which touches c_1 externally and also the two tangents P A and P B is 2sqrt(3)-3 (b) 2sqrt(2)-1 2sqrt(2)-1 (d) 1

If in a right-angled triangle A B C , the hypotenuse A B=p ,then vec(AB).vec(AC)+ vec(BC). vec(BA)+ vec(CA).vec(CB) is equal to 2p^2 b. (p^2)/2 c. p^2 d. none of these

If vec a , vec ba n d vec c are unit coplanar vectors, then the scalar triple product [2 vec a- vec b2 vec b- vec c2 vec c- vec a] is a. 0 b. 1 c. -sqrt(3) d. sqrt(3)

If vec a , vec ba n d vec c are unit vectors satisfying | vec a- vec b|^2+| vec b- vec c|^2+| vec c- vec a|^2=9, then |2 vec a+5 vec b+5 vec c| is.

If A ,B ,C ,D are four distinct point in space such that A B is not perpendicular to C D and satisfies vec A Bdot vec C D=k(| vec A D|^2+| vec B C|^2-| vec A C|^2-| vec B D|^2), then find the value of kdot

Let vec a=2i+j-2ka n d vec b=i+jdot If vec c is a vector such that vec adot vec c=| vec c|,| vec c- vec a|=2sqrt(2) between vec axx vec b and vec ci s30^0,t h e n|( vec axx vec b)xx vec c| I equal to a. 2//3 b. 3//2"" c. 2 d. 3

If vec r_1, vec r_2, vec r_3 are the position vectors of the collinear points and scalar p a n d q exist such that vec r_3=p vec r_1+q vec r_2, then show that p+q=1.