# How to solve 3-variable planar vector equilibrium problems?

While it's quite straightforward to solve problems with multiple planar vectors in equilibrium for 2 variables, I'm having issues with those where I'm asked to minimize or maximize a variable along with two others. For example: We are asked to find:

• $$\theta$$ such that $$F_1$$ is as small as possible
• $$F_1$$ and $$F_2$$ such that our resulting vector $$F_R = (3 kN, \angle 0°)$$

I've gone ahead and established the equilibrium for the $$x$$ and $$y$$ axes but I'm at a loss as to how to continue from there, because I don't know $$F_2$$ to solve for $$\theta$$, and because I'm assuming this can be done without calculus, although I might be wrong.

$$\sum F_x=0=-5\cos(30°)+F_2 \cos(20°)+F_1 \cos(\theta) -7\cdot4/5$$ $$\sum F_y=0=5\sin(30°)+F_2 \sin(20°)+F_1 \sin(\theta) -7\cdot3/5$$

• Could elaborate more on the question? – Manvendra Somvanshi Apr 29 at 0:08
• How do you mean? – Ariel Arévalo Apr 29 at 3:48
• You have a third constraint, don't forget $F_R$ in your analysis, this will help with the relationship between $F_1$ and $F_2$. – Triatticus Apr 29 at 8:16
• That helps me with the second question but what about minimizing $F_1$? – Ariel Arévalo Apr 30 at 1:21