Three vectors a, b and c are such that | vector a|= 1| vector b|= 2| vector c| = 4 and (vector a+vector b+vector c)=0, then what is the value of 4 (vector a) . (vector b)+ 3 (vector b) . (vector c) + 3 (vector c). (vector a)?

Since the resultant of three vectors are zero, they three must be forming a triangle.(where magnitude of two vectors is the length of the third side) Let the three angles be alpha, beta and gamma. If A and B are two vectors and theta is the exterior angle (actually the angle between them when placed tail to tail, but here they are placed tail-head overlap)between them, the resultant`s magnitude will be sqrt(A^2 + B^2 +2|A||B|cos(theta)). By this you can find the cosine of three angles.(exterior angles). Then with the formula for dot product A.B = |A||B|cos(theta). With that you can get the answer. Let me know if you need further help.