Editing Center of mass (section)

===In three dimensions===
An experimental method to locate the three-dimensional coordinates of the center of mass begins by supporting the object at three points and measuring the forces, '''F'''<sub>1</sub>, '''F'''<sub>2</sub>, and '''F'''<sub>3</sub> that resist the weight of the object, <math>\mathbf{W} = -W\mathbf{\hat{k}}</math> (<math>\mathbf{\hat{k}}</math> is the unit vector in the vertical direction).  Let '''r'''<sub>1</sub>, '''r'''<sub>2</sub>, and '''r'''<sub>3</sub> be the position coordinates of the support points, then the coordinates '''R''' of the center of mass satisfy the condition that the resultant torque is zero,
<math display="block">\mathbf{T} = (\mathbf{r}_1 - \mathbf{R}) \times \mathbf{F}_1 + (\mathbf{r}_2 - \mathbf{R}) \times \mathbf{F}_2 + (\mathbf{r}_3 - \mathbf{R}) \times \mathbf{F}_3 = 0,</math>
or
<math display="block">\mathbf{R} \times \left(-W\mathbf{\hat{k}}\right) = \mathbf{r}_1 \times \mathbf{F}_1 + \mathbf{r}_2 \times \mathbf{F}_2 + \mathbf{r}_3 \times \mathbf{F}_3. </math>

This equation yields the coordinates of the center of mass '''R'''* in the horizontal plane as,
<math display="block"> \mathbf{R}^* = -\frac{1}{W} \mathbf{\hat{k}} \times (\mathbf{r}_1 \times \mathbf{F}_1 + \mathbf{r}_2 \times\mathbf{F}_2 + \mathbf{r}_3 \times \mathbf{F}_3).</math>

The center of mass lies on the vertical line '''L''', given by
<math display="block"> \mathbf{L}(t) = \mathbf{R}^* + t\mathbf{\hat{k}}.</math>

The three-dimensional coordinates of the center of mass are determined by performing this experiment twice with the object positioned so that these forces are measured for two different horizontal planes through the object.  The center of mass will be the intersection of the two lines '''L'''<sub>1</sub> and '''L'''<sub>2</sub> obtained from the two experiments.