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

If O( vec0 ) is the circumcentre and O' the orthocentre of a triangle ABC, then prove that i. vec(OA)+vec(OB)+vec(OC)=vec(OO') ii. vec(O'A)+vec(O'B)+vec(O'C)=2vec(O'O) iii. vec(AO')+vec(O'B)+vec(O'C)=2 vec(AO)=vec(AP) where AP is the diameter through A of the circumcircle.

If D is the midpoint of the side BC of a triangle ABC, prove that vec(AB)+vec(AC)=2vec(AD)

If D is the midpoint of the side AB of a triagle ABC prove that vec(BC)+vec(AC)=-2vec(CD)

If D and E, are the midpoints of the sides AB and AC of a triangle ABC, prove that vec(BE) +vec(DC)=(3)/(2)vec(BC).

The position vectors of the vertices of a triangle are vec(i) +2 vec(j) +3 vec(k) , 3 vec(i) -4 vec(j) +5 vec (k) and -2vec (i) +3 vec (j) - 7 vec (k) Find the perimeter of a triangle .

If ABCDE is a pentagon then prove that vec(AB)+vec(AE)+vec(BC)+vec(DC)+vec(ED)+vec(AC)=3vec(AC)

prove that [vec(a)-vec(b),vec(b)-vec(c)vec(c)-vec(a)]=0

Orthocenter of an equilateral triangle ABC is the origin O. If vec(OA)=veca, vec(OB)=vecb, vec(OC)=vecc , then vec(AB)+2vec(BC)+3vec(CA)=

(Apollonius theorem): If D is the midpoint of the side BC of a triangle ABC, then show by vector method that |vec(AB)|^(2) + |vec(AC)|^(2) = 2(|vec(AD)|^(2)+ |vec(BD)|^(2)).

Let ABC be a triangle whose centroid is G, orhtocentre is H and circumcentre is the origin 'O'. If D is any point in the plane of the triangle such that no three of O, A, C and D are collinear satisfying the relation vec(AD) + vec(BD) + vec(CH ) + 3 vec(HG)= lamda vec(HD) , then what is the value of the scalar 'lamda' ?