Let `A B C D` be a parallelogram whose diagonals intersect at `P` and let `O` be the origin. Then prove that ` vec O A+ vec O B+ vec O C+ vec O D=4 vec O Pdot`

Let A B C D be a p[arallelogram whose diagonals intersect at P and let O be the origin. Then prove that vec O A+ vec O B+ vec O C+ vec O D=4 vec O Pdot

If A B C D is a rhombus whose diagonals cut at the origin O , then proved that vec O A+ vec O B+ vec O C+ vec O D =0

Let PQRS be a parallelogram whose diagonals PR and QS intersect at O'. If O is the origin, then (vec(OP)+vec(OQ)+vec(OR)+vec(OS)) is equal to -

If vec a+ vec b+ vec c = vec 0 then prove that vec axx vec b= vec bxx vec c = vec cxxvec a .

Let vec a , vec b ,and vec c be any three vectors, then prove that [ vec axx vec b vec bxx vec c vec cxx vec a ]= [vec a vec b vec c]^2

A B C D is a quadrilateral. E is the point of intersection of the line joining the midpoints of the opposite sides. If O is any point and vec O A+ vec O B+ vec O C+ vec O D=x vec O E ,t h e nx is equal to a. 3 b. 9 c. 7 d. 4

Prove that [[ vec a+ vec b, vec b+ vec c, vec c+ vec a]]=2[ [vec a, vec b, vec c]]dot

Let O A C B be a parallelogram with O at the origin and O C a diagonal. Let D be the midpoint of O Adot using vector methods prove that B Da n dC O intersect in the same ratio. Determine this ratio.

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

Let vec a , vec b ,a n d vec c be non-coplanar vectors and let the equation vec a^' , vec b^' , vec c ' are reciprocal system of vector vec a , vec b , vec c , then prove that vec axx vec a^'+ vec bxx vec b^'+ vec cxx vec c ' is a null vector.