### Vector Addition and Resultant ForceTwo forces \(\mathbf{F}_1 = 112 \left( \frac{\mathbf{i} - \mathbf{j}}{\sqrt{2}} \right)\) N and \(\mathbf{F}_2 = 388 \left( \frac{\mathbf{i} - \mathbf{j}}{\sqrt{2}} \right)\) N act on an object.---#### Questions:1. **What is the resultant force in terms of the force vectors \(\mathbf{F}_1\) and \(\mathbf{F}_2\)?**2. **What is the magnitude of an equal and opposite force, \(\mathbf{F}_3\), which balances the first two forces?**#### Input Field:\(\mathbf{F}_3\): [Input Box]#### Instructions:Use the calculator below, if needed, to input your final answer. You can switch between degrees and radians for trigonometric functions.### Calculator Interface:- **Functions:** - Trigonometric functions (sin, cos, tan, etc.) - Hyperbolic functions (sinh, cosh, tanh, etc.) - Inverse trigonometric functions (asin, acos, atan, etc.)- **Common Constants:** - Pi (\(\pi\)) - Euler's number (\(e\))- **Basic Operations:** - Addition (+), Subtraction (-), Multiplication (*), Division (/)- **Special Functions:** - Degrees/Radians toggle - Square root (\(\sqrt{\cdot}\)) - Exponential (\(E)\)---### Further Analysis:**Given vectors:**\[\mathbf{F}_1 = c_1 \left( \frac{\mathbf{i} - \mathbf{j} + \mathbf{k}}{\sqrt{n}} \right) \text{ N} \quad \text{and} \quad \mathbf{F}_2 = c_2 \left( \frac{\mathbf{i} - \mathbf{j} + \mathbf{k}}{\sqrt{n}} \right) \text{ N}\]**Finding the Normalizing Term \(n\):**What will \(n\) be for the two vectors \(\mathbf{F}_1\) and \(\mathbf{F}_2\)?**Magnitude of Resultant Vector \( \mathbf

Name: ### Vector Addition and Resultant ForceTwo forces \(\mathbf{F}_1 = 11
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Description: ### Vector Addition and Resultant ForceTwo forces \(\mathbf{F}_1 = 112 \left( \frac{\mathbf{i} - \mathbf{j}}{\sqrt{2}} \right)\) N and \(\mathbf{F}_2 = 388 \left( \frac{\mathbf{i} - \mathbf{j}}{\sqrt{2}} \right)\) N act on an object.---#### Question

Answered: Image /qna-images/answer/48cc4dad-e309-4bbc-b36a-c362434c40fe.jpg

Answered: F3² || Two forces of F₁ = 112. (i-j) √2…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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### Vector Addition and Resultant Force

Two forces \(\mathbf{F}_1 = 112 \left( \frac{\mathbf{i} - \mathbf{j}}{\sqrt{2}} \right)\) N and \(\mathbf{F}_2 = 388 \left( \frac{\mathbf{i} - \mathbf{j}}{\sqrt{2}} \right)\) N act on an object.

---

#### Questions:
1. **What is the resultant force in terms of the force vectors \(\mathbf{F}_1\) and \(\mathbf{F}_2\)?**
2. **What is the magnitude of an equal and opposite force, \(\mathbf{F}_3\), which balances the first two forces?**

#### Input Field:
\(\mathbf{F}_3\): [Input Box]

#### Instructions:
Use the calculator below, if needed, to input your final answer. You can switch between degrees and radians for trigonometric functions.

### Calculator Interface:
- **Functions:**
- Trigonometric functions (sin, cos, tan, etc.)
- Hyperbolic functions (sinh, cosh, tanh, etc.)
- Inverse trigonometric functions (asin, acos, atan, etc.)
- **Common Constants:**
- Pi (\(\pi\))
- Euler's number (\(e\))
- **Basic Operations:**
- Addition (+), Subtraction (-), Multiplication (*), Division (/)
- **Special Functions:**
- Degrees/Radians toggle
- Square root (\(\sqrt{\cdot}\))
- Exponential (\(E)\)

---

### Further Analysis:

**Given vectors:**
\[
\mathbf{F}_1 = c_1 \left( \frac{\mathbf{i} - \mathbf{j} + \mathbf{k}}{\sqrt{n}} \right) \text{ N} \quad \text{and} \quad \mathbf{F}_2 = c_2 \left( \frac{\mathbf{i} - \mathbf{j} + \mathbf{k}}{\sqrt{n}} \right) \text{ N}
\]

**Finding the Normalizing Term \(n\):**
What will \(n\) be for the two vectors \(\mathbf{F}_1\) and \(\mathbf{F}_2\)?

**Magnitude of Resultant Vector \( \mathbf

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Follow-up Question

What is the magnitude of the resultant vector in terms of the constants, given then normalizing term n?

### Vector Calculation in 3 Dimensions

Given the normalizing term \( n \) found in the previous part in 3 dimensions \( \left(\hat{i}, \hat{j}, \hat{k}\right) \), what will the magnitude of the resultant vector \( \left|\vec{F}_{res}\right| \) be in terms of the constants \( c_1 \) and \( c_2 \)?

The vectors in question are given as:

\[
\vec{F}_1 = c_1 \left(\frac{\hat{i} - \hat{j} + \hat{k}}{\sqrt{n}}\right) \text{ N} \quad \text{and} \quad \vec{F}_2 = c_2 \left(\frac{\hat{i} - \hat{j} + \hat{k}}{\sqrt{n}}\right).
\]

### Explanation of the Formula
This equation shows how vectors are normalized and expressed in terms of unit vectors \( (\hat{i}, \hat{j}, \hat{k}) \) across three dimensions. \( c_1 \) and \( c_2 \) are constants that scale the vectors. The term \( \frac{1}{\sqrt{n}} \) normalizes the vector in question.

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Follow-up Question

How would the last part of this problem be solved (finding the magnitude of the resultant vector in terms of the two constants given the normalizing term n)?

## Forces and Resultant Force Calculation

### Problem Statement:
Two forces are acting on an object:
- \( \mathbf{F}_1 = 112 \left( \frac{(i - j)}{\sqrt{2}} \right) \) N
- \( \mathbf{F}_2 = 38 \left( \frac{(i - j)}{\sqrt{2}} \right) \) N

### Questions:
1. What is the resultant force in terms of the force vectors \( \mathbf{F}_1 \) and \( \mathbf{F}_2 \)?
2. What is the magnitude of an equal and opposite force, \( \mathbf{F}_3 \), which balances the first two forces?

The general formula provided for the resultant force \( \mathbf{F}_3 \) is:
\[ \mathbf{F}_3 = \]

### Trigonometric Function Evaluation:
Here we have a trigonometric function evaluation table with input values in both degrees and radians.

| | sin() | cos() | tan() | csc() | sec() | cot() |
|-------|-----------|-----------|-----------|-----------|----------|-----------|
| 30 | | | | | | |
| 45 | | | | | | |
| 60 | | | | | | |
| 90 | | | | | | |

Options to choose Degrees or Radians: ![Degree Radian Toggle](#)

### Vector Analysis:
Given your observations from parts a and b, if working with vectors in 3 dimensions \( (i, j, k) \), what will the normalizing term **c** be for the two vectors \( \mathbf{F}_1 \) and \( \mathbf{F}_2 \):
\[ \mathbf{F}_1 = c( \frac{(i - j + k)}{\sqrt{n}} ) \quad \text{and} \quad \mathbf{F}_2 = c ( \frac{(i - j + k)}{\sqrt{n}} ) \]

Given the normalizing term **c** found in the previous part in 3 dimensions \( (i, j, k) \), what will the magnitude of the resultant vector \( \left \

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Follow-up Question

Is normalizing in this case creating a unit vector or a vector perpendicular to F1 and F2?

**Problem Statement:**

Given your observations from parts a and b, if working with vectors in 3 dimensions \( \left(\mathbf{i}, \mathbf{j}, \mathbf{k}\right) \), what will the normalizing term \( k \) be for the two vectors \(\mathbf{F}_1\) and \(\mathbf{F}_2\)?

**Expression for Vectors:**
\[
\mathbf{F}_1 = c_1 \frac{( \mathbf{i} - \mathbf{j} + \mathbf{k} )}{\sqrt{n}} \text{ N and } \mathbf{F}_2 = c_2 \frac{( \mathbf{i} - \mathbf{j} + \mathbf{k} )}{\sqrt{n}} \text{ N.}
\]

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