Summing Unequal Magnitude Vectors to Reach Zero

[ThomasMagnus]

Hi,

I'm looking for some help on how the sum of a certain number of vectors can equal zero. I know that the sum of 2 vectors with equal magnitudes but opposite directions will equal zero; 2 vectors of unequal magnitude can never have a sum equal zero; and that three vectors of unequal magnitude can have a sum of zero if they form a closed triangle.

Three vectors of unequal magnitude can have a sum of zero if they form a closed triangle.

For this to be true, does the final vector have to point to the origin, or is it just a triangle anywhere?

What about four vectors? Can four vectors ever have a sum of zero if they have equal or unequal magnitude?

Here is a picture of a few vectors that I think have a sum of zero. Correct me if I am wrong.

Thanks =)

 

[Drakkith]

I don't know too much about vectors, but I believe that if they form a closed shape then they will equal zero. Don't take my word on it though.

 

[ThomasMagnus]

That's what I was thinking also. Can anyone confirm this?

 

[pongo38]

Confirmed but... The vertical components add up to zero. And so do the horizontals. (Actually, any two non-parallel directions will do). However, it's not the only requirement for equilibrium. To satisfy equilibrium, the algebraic sum of the moments about ANY point must also be zero. Consider a square thing with a south facing force at the top left corner, and a north facing force of the same magnitude at the bottom right corner. The vector diagram closes, but the object will spin anticlockwise, and js therefore not in static equilibrium.

 

[chiro]

For the case you're talking about think of a polygon in n sides.

In this case the polygon can be convex or concave: there is no restriction on the orientation or length of the edges just as long as the shape is in fact a polygon (edges connect at vertices with one vertex being the origin.