G is a point inside the plane of the triangle ABC, if `vec(GA)+vec(GB)+vec(GC)=vec(0)`, then show that G is the centroid of the triangle ABC.

If G be the centroid of the triangle ABC, then prove that vec(GA)+vec(GB)+vec(GC)=vec(0).

If vec(AO)+vec(OB)=vec(BO)+vec(OC) , show that the points A, B and C are collinear.

ABC is a triangle and P any point on BC. If vec(PQ) is the sum of vec(AP)+vec(PB)+vec(PC) , show that ABQC is a parallelogram and Q, therefore, is a fixed point.

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

D, E, F are the midpoints of the sides bar(BC), bar(CA) " and " bar(AB) respectively of the triangle ABC. If P is any point in the plane of the triangle, show that vec(PA)+vec(PB)+vec(PC)=vec(PD)+vec(PE)+vec(PF) .

If vec(a) = vec(0) or vec(b) = vec(0) then vec(a).vec(b) = 0 , show by an example that the converse of this statement is not always true .

For the vector vec(a) and vec (b) if |vec(a) + vec(b)| = |vec(a) - vec(b)| , show that vec(a) and vec (b) are perpendicular

If G is the centroid of triangle A B C ,t h e n vec G A+ vec G B+ vec G C is equal to a. vec0 b. 3 vec G A c. 3 vec G B d. 3 vec G C

Let vec a , vec b , vec c , vec d be the position vectors of the four distinct points A , B , C , Ddot If vec b- vec a= vec c- vec d , then show that A B C D is parallelogram.

If vec(a),vec(b),vec (c ) are non - coplanar vectors vec (r) . vec(a) = vecr . vec(b) = vecr.vec(c ) = 0 , show that vec (r ) is a zero vector .