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`

Let D ,Ea n dF be the middle points of the sides B C ,C Aa n dA B , respectively of a triangle A B Cdot Then prove that vec A D+ vec B E+ vec C F= vec0 .

Let vectors vec a , vec b , vec c ,a n d vec d be such that ( vec axx vec b)xx( vec cxx vec d)=0. Let P_1a n dP_2 be planes determined by the pair of vectors vec a , vec b ,a n d vec c , vec d , respectively. Then the angle between P_1a n dP_2 is a. 0 b. pi//4 c. pi//3 d. pi//2

In triangle A B C ,points D , Ea n dF are taken on the sides B C ,C Aa n dA B , respectively, such that (B D)/(D C)=(C E)/(E A)=(A F)/(F B)=ndot Prove that triangle D E F=(n^2-n+1)/((n+1)^2) triangle(A B C)dot

The position vectors of the vertices A ,Ba n dC of a triangle are three unit vectors vec a , vec b ,a n d vec c , respectively. A vector vec d is such that vec d ⋅ vec a = vec d ⋅ vec b = vec d ⋅ vec c a n d vec d=lambda( vec b+ vec c)dot Then triangle A B C is a. acute angled b. obtuse angled c. right angled d. none of these

Given the vectors vec A , vec B ,a n d vec C form a triangle such that vec A= vec B+ vec Cdot find a ,b ,c ,a n dd such that the area of the triangle is 5sqrt6 where vec A=a hat i+b hat j+c hat k vec B=d hat i+3 hat j+4 hat k vec C=3 hat i+ hat j-2 hat k

If vec a , vec b , vec c , vec d are the position vector of point A , B , C and D , respectively referred to the same origin O such that no three of these point are collinear and vec a + vec c = vec b + vec d , than prove that quadrilateral A B C D is a parallelogram.

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 three vectors in the plane of vec a , vec b ; vec b , vec c ; vec c , vec a , respectively. Then a. vec x. vec d=-1 b. vec y. vec d=1 c. vec z. vec d=0 d. vec r. vec d=0,w h e r e vec r=lambda vec x+mu vec y+delta vec z

Let vec a , vec b a n d vec c be three vectors having magnitudes 1, 5and 3, respectively, such that the angel between vec aa n d vec bi stheta and vec axx( vec axx vec b)=c . Then t a ntheta is equal to a. 0 b. 2/3 c. 3/5 d. 3/4

A ,B ,C and D have position vectors vec a , vec b , vec c and vec d , respectively, such that vec a- vec b=2( vec d- vec c)dot Then a. A Ba n dC D bisect each other b. B Da n dA C bisect each other c. A Ba n dC D trisect each other d. B Da n dA C trisect each other

If ( vec axx vec b)xx( vec bxx vec c)= vec b ,w h e r e vec a , vec b ,a n d vec c are nonzero vectors, then 1. vec a , vec b ,a n d vec c can be coplanar 2. vec a , vec b ,a n d vec c must be coplanar 3. vec a , vec b ,a n d vec c cannot be coplanar 4.none of these