**Vector Components to Balance Forces****Problem Statement:** 1. Use vector components to find the magnitude and direction of a force vector needed to balance the forces of 400 N @ 145 degrees and 400 N @ 260 degrees.**Approach:**To solve this problem, we will decompose the given forces into their respective vector components, sum the components, and determine the resultant force vector that balances the system.**Step-by-Step Solution:**1. **Force Decomposition:** For each force \( F \): \[ F_x = F \cdot \cos(\theta) \] \[ F_y = F \cdot \sin(\theta) \] where \( F \) is the magnitude and \( \theta \) is the direction of the force.2. **Calculate Components of Given Forces:** For the force 400 N @ 145 degrees: \[ F_{1x} = 400 \cdot \cos(145^\circ) = 400 \cdot (-0.819) \approx -327.6 \, N \] \[ F_{1y} = 400 \cdot \sin(145^\circ) = 400 \cdot (0.5736) \approx 229.4 \, N \] For the force 400 N @ 260 degrees: \[ F_{2x} = 400 \cdot \cos(260^\circ) = 400 \cdot (-0.1736) \approx -69.44 \, N \] \[ F_{2y} = 400 \cdot \sin(260^\circ) = 400 \cdot (-0.9848) \approx -393.92 \, N \]3. **Sum of the Components:** Sum of all \( x \)-components: \[ \Sigma F_x = F_{1x} + F_{2x} = -327.6 \, N + (-69.44 \, N) = -397.04 \, N \] Sum of all \( y \)-components: \[ \Sigma F_y = F_{1y} + F_{2y} = 229.4 \, N + (-

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Description: **Vector Components to Balance Forces****Problem Statement:** 1. Use vector components to find the magnitude and direction of a force vector needed to balance the forces of 400 N @ 145 degrees and 400 N @ 260 degrees.**Approach:**To solve this probl

Given forces:. 400 N at 145 degree. 400 N at 260 degree.

Answered: 3. Use vector components to find the…

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3. Use vector components to find the magnitude and direction of a force vector needed to balance the forces of 400 N @ 145 degrees and 400 N @ 260 degrees.

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Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

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Question

**Vector Components to Balance Forces**

**Problem Statement:**

1. Use vector components to find the magnitude and direction of a force vector needed to balance the forces of 400 N @ 145 degrees and 400 N @ 260 degrees.

**Approach:**

To solve this problem, we will decompose the given forces into their respective vector components, sum the components, and determine the resultant force vector that balances the system.

**Step-by-Step Solution:**

1. **Force Decomposition:**

For each force \( F \):

\[
F_x = F \cdot \cos(\theta)
\]
\[
F_y = F \cdot \sin(\theta)
\]

where \( F \) is the magnitude and \( \theta \) is the direction of the force.

2. **Calculate Components of Given Forces:**

For the force 400 N @ 145 degrees:
\[
F_{1x} = 400 \cdot \cos(145^\circ) = 400 \cdot (-0.819) \approx -327.6 \, N
\]
\[
F_{1y} = 400 \cdot \sin(145^\circ) = 400 \cdot (0.5736) \approx 229.4 \, N
\]

For the force 400 N @ 260 degrees:
\[
F_{2x} = 400 \cdot \cos(260^\circ) = 400 \cdot (-0.1736) \approx -69.44 \, N
\]
\[
F_{2y} = 400 \cdot \sin(260^\circ) = 400 \cdot (-0.9848) \approx -393.92 \, N
\]

3. **Sum of the Components:**

Sum of all \( x \)-components:
\[
\Sigma F_x = F_{1x} + F_{2x} = -327.6 \, N + (-69.44 \, N) = -397.04 \, N
\]

Sum of all \( y \)-components:
\[
\Sigma F_y = F_{1y} + F_{2y} = 229.4 \, N + (-

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