Prove that `3vec(OD)+vec(DA)+vec(DB)+vec(DC)` is equal to `vec(OA)+vec(OB)+vec(OC)`.

if A,B,C,D and E are five coplanar points, then vec(DA) + vec( DB) + vec(DC ) + vec( AE) + vec( BE) + vec( CE) is equal to :

Prove that [vec(a)+vec(b),vec(b)+vec( c) , vec( c) +vec(a)] =2 [vec(a),vec(b),vec(c )] .

If vec(a), vec(b)" and "vec(c) are three unit vectors such that vec(a)+vec(b)+vec(c)=vec(O) , find the value of vec(a).vec(b)+vec(b).vec(c)+vec(c).vec(a) .

If vec(a), vec(b), vec(c ) are unit vectors such that vec(a) + vec(b) + vec(c )= vec(0), " then " vec(a).vec(b) + vec(b).vec(c ) + vec(c ).vec(a) =

If vec(a), vec(b),vec(c ) are three vectors such that |vec(a)| = 5, |vec(b)| = 12 and |vec(c )| = 13 and vec(a) + vec(b) + vec(c ) = vec(0) , find the value of vec(a).vec(b) + vec(b).vec(c )+vec(c ).vec(a) .

ABCDE is a pentagon. Prove that the resultant of forces vec (AB), vec(AE), vec(BC), vec(DC), vec(ED) and vec(AC) is 3vec(AC) .

Prove that [vec(a),vec(b),vec( c) +vec(d)]=[vec(a),vec(b),vec( c)] +[vec(a),vec(b),vec(d)] .

Let A and B be points with position vectors veca and vecb with respect to origin O . If the point C on OA is such that 2vec(AC)=vec(CO), vec(CD) is parallel to vec(OB) and |vec(CD)|=3|vec(OB)| then vec(AD) is

If vec(a) = 3vec(i) - 2vec(j) + 2vec(i), vec(b) = 6vec(i) + 4vec(j) - 2vec(k), vec(c ) = 3hat(i) - 2hat(j) - 4hat(k) , Then vec(a).(vec(b) xx vec(c )) is