# As a system of linear equations show lines of action of 3 co-planar forces in equilibrium meet at a point.

I have multiple ways of showing that the lines of action of 3 co-planar forces in static equilibrium meet at a point. But they combine physical arguments and math. It seems like there should be some purely mathematical way of proving the claim, treating it as a system of equations having a unique solution.

Is there a way to express this as a uniquely determined system of linear equations?

Here are the equations of equilibrium corresponding to the illustration, where $$\vec{r}_1=\mathfrak{p}_1-\mathscr{O},$$ etc. The heavy gray line is a rigid rod in equilibrium under the applied forces.

\begin{aligned} \vec{0}&=\vec{f}_{1}+\vec{f}_{2}+\vec{f}_{3}\\ \vec{0}&=\vec{r}_{1}\times\vec{f}_{1}+\vec{r}_{2}\times\vec{f}_{2}+\vec{r}_{3}\times\vec{f}_{3}\\ \vec{0}&=\left(\vec{r}_{1}-\vec{p}\right)\times\vec{f}_{1}+\left(\vec{r}_{2}-\vec{p}\right)\times\vec{f}_{2}+\left(\vec{r}_{3}-\vec{p}\right)\times\vec{f}_{3} \end{aligned}

The equation of a line through $$r_i$$ in the direction $$f_i$$ is $$(r-r_i)\times f_i=0$$ which we can rewrite as $$r\times f_i=r_i\times f_i$$. Now assuming $$f_1$$ and $$f_2$$ are not parallel, they intersect at some point $$p$$, so $$p\times f_1=r_1\times f_1$$ and $$p\times f_2=r_2\times f_2$$ which gives that
$$p\times(f_1+f_2)=r_1\times f_1+r_2\times f_2.$$ Now add $$p\times f_3$$ to both sides, to give $$0=r_1\times f_1+r_2\times f_2+p\times f_3$$ by the fact that the forces are in equilibrium (your first equation). But using the fact that the moments are in equilibrium (your second equation) gives that $$r_1\times f_1+r_2\times f_2+r_3\times f_3=r_1\times f_1+r_2\times f_2+p\times f_3$$ so $$r_3\times f_3=p\times f_3$$ which exactly says that $$p$$ (the intersection of the first two lines) also lies on the third line, thus the lines all intersect at a common point.
• The hook I was missing was the equation of a line in the form $(r-r_i)\times f_i=0$. I'm certain I've encountered it before, but its practical utility was lost on me until now. – Steven Thomas Hatton Apr 8 at 17:48