MHB What Does the Encircled Equation Mean?

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The discussion centers on understanding the encircled equation that equates the $\hat{i}$ and $\hat{j}$ components of forces. Participants clarify that the resultant force $\mathbf F_R$ is zero, meaning both horizontal and vertical force components must also equal zero. This leads to the conclusion that the forces acting on the system are balanced, resulting in no net force. One user questions why the components become zero, suggesting they cancel each other out. The explanation confirms that the problem states the resultant force is zero, necessitating the individual components to also be zero.
Drain Brain
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Hello! :)

I just want to ask how, did the encircled portion come about?

It says that it equates the $\hat{i}$ and $\hat{j}$ components.

but when I tried that, this is what I get

$0.5447F_{1}=F_{3}(\sin(\theta)-\cos(\theta))$ ---> This expression doesn't ring a bell. It doesn't make any sense to me.
 

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Drain Brain said:
Hello! :)

I just want to ask how, did the encircled portion come about?

It says that it equates the $\hat{i}$ and $\hat{j}$ components.

but when I tried that, this is what I get

$0.5447F_{1}=F_{3}(\sin(\theta)-\cos(\theta))$ ---> This expression doesn't ring a bell. It doesn't make any sense to me.

Hi Drain Brain,

The $\mathbf F_R$ on the left hand side is really:
$$\mathbf F_R = 0\cdot \boldsymbol{\hat\imath} + 0 \cdot \boldsymbol{\hat\jmath}$$
Since the $\boldsymbol{\hat\imath}$ component is completely independent from the $\boldsymbol{\hat\jmath}$, we match them left and right.

Put otherwise, we separate the equation in:
- Sum of the horizontal forces is zero
- Sum of the vertical forces is zero
 
I like Serena said:
Hi Drain Brain,

The $\mathbf F_R$ on the left hand side is really:
$$\mathbf F_R = 0\cdot \boldsymbol{\hat\imath} + 0 \cdot \boldsymbol{\hat\jmath}$$
Since the $\boldsymbol{\hat\imath}$ component is completely independent from the $\boldsymbol{\hat\jmath}$, we match them left and right.

Put otherwise, we separate the equation in:
- Sum of the horizontal forces is zero
- Sum of the vertical forces is zero

Hi, I Like Serena!

Can you tell me why the i and j components become 0?

What I'm thinking about the problem is that the components of $F_{R}$ are equal in magnitude but opposite in direction that's why they cancel each other and produce a resultant of 0. Is my line of thinking correct? If not please explain to me why the components became both 0. Thanks!
 
Drain Brain said:
Hi, I Like Serena!

Can you tell me why the i and j components become 0?

What I'm thinking about the problem is that the components of $F_{R}$ are equal in magnitude but opposite in direction that's why they cancel each other and produce a resultant of 0. Is my line of thinking correct? If not please explain to me why the components became both 0. Thanks!

It's because the problem statement says:

The three concurrent forces acting on the screw eye produce a resultant force of $\mathbf F_R = \mathbf 0$.
 
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