Calculating torque in 3D?

Question

Say you have a sphere, and you have several torque vectors acting on it, all at different points. Say you have the vector (6i + 3j + 5k) originating from point A, and the vector (3i + 1j + 9k) originating at point B, and (7i + 2j + 9k) acting on point C.
Summing the vectors gives you (16i + 6j + 23k) which is the resultant moment/torque vector. But at what point does the moment act on - A, B, or C?
The point acts on has to matter right? I mean if you think of the moment vector as an axis the sphere revolves around, placing it in the center of the sphere and rotating the sphere around that is clearly different from placing it at the far left of the sphere and rotating it around that.

John Alexiou · Accepted Answer

So you know about how to get the effective moment of all the forces

$$  vec{M} = sum_{i} vec{r}_i times vec{F}_i $$

and the total forces

$$ vec{F} = sum_i vec{F}_i $$

To get the location where the moments balance out (the line of action of the combined force) you do the following

$$ vec{r} = -frac{vec{M} times vec{F}}{vec{F} cdot vec{F}} $$

for example a force $vec{F}=(1,0,0)$ located at $vec{r}=(0,y,z)$ creates a torque of $vec{M}=(0,z,-y)$. To recover the location of the force do $$r = -frac{ (0,z,-y) times (1,0,0) }{(1,0,0)cdot (1,0,0)} =- frac{(0,-y,-z)}{1} = (0,y,z) $$

Calculating torque in 3D?

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