# 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:

• $$\theta$$ 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.
$$\sum F_x=0=-5\cos(30°)+F_2 \cos(20°)+F_1 \cos(\theta) -7\cdot4/5$$ $$\sum F_y=0=5\sin(30°)+F_2 \sin(20°)+F_1 \sin(\theta) -7\cdot3/5$$
• 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
The text passage discusses a problem involving the equilibrium of planar vectors with multiple variables, presenting two specific questions and some equations to set up the problem. It also includes a comment suggesting the use of an additional constraint.
multiple_questions
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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:

• $$\theta$$ 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.
$$\sum F_x=0=-5\cos(30°)+F_2 \cos(20°)+F_1 \cos(\theta) -7\cdot4/5$$ $$\sum F_y=0=5\sin(30°)+F_2 \sin(20°)+F_1 \sin(\theta) -7\cdot3/5$$
• 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
user
The text passage discusses a problem involving the equilibrium of planar vectors with multiple variables, presenting two specific questions and some equations to set up the problem. It also includes a comment suggesting the use of an additional constraint.

TYPE: MULTIPLE_QUESTIONS