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

Let D, E, F are the midpoints of the sides bar(BC), bar(CA) " and " bar(AB) of the triangle ABC. Prove that vec(AD)+vec(BE)+vec(CF)=vec(0) .

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 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

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) .

Let vec(a),vec(b),vec (c ) be the positions vectors of the vertices of a triangle , prove that the area of the triangle is 1/2| vec(a) xx vec(b) + vec(b) xx vec(c)+ vec(c)xx vec(a)|

Three non-zero vectors vec(a), vec(b), vec(c) are such that no two of them are collinear. If the vector (vec(a)+vec(b)) is collinear with the vector vec(c) and the vector (vec(b)+vec(c)) is collinear with vector vec(a) , then prove that (vec(a)+vec(b)+vec(c)) is a null vector.

If vec a= vec p+ vec q , vec pxx vec b=0a n d vec qdot vec b=0, then prove that ( vec bxx( vec axx vec b))/( vec bdot vec b)= vec qdot

If vec a and vec b are two vectors, then prove that ( vec axx vec b)^2=|[vec a.vec a,vec a.vec b],[vec b.vec a,vec b.vec b]| .

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

If vec a , vec b , vec ca n d vec d are the position vectors of the vertices of a cyclic quadrilateral A B C D , prove that (| vec axx vec b+ vec bxx vec d+ vec d xx vec a|)/((vec b- vec a).(vec d- vec a)) + (| vec bxx vec c+ vec cxx vec d+ vec d xx vec b|)/((vec b- vec c).( vec d- vec c))=0dot