### Displacement Vectors ProblemConsider the following three displacement vectors:1. **Vector \( \vec{A} \)** has a magnitude of 3.30 km and a direction that makes an angle of 35.0° north of west.2. **Vector \( \vec{B} \)** has a magnitude of 7.40 km and a direction that makes an angle of 12.0° south of east.3. **Vector \( \vec{C} \)** has a magnitude of 4.30 km and a direction that makes an angle of 47.0° south of east.Determine the magnitude of the vector \( \vec{Z} = \vec{C} - \vec{B} + \vec{A} \).**Answer:** \[ \boxed{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{km} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \]### Detailed Explanation of Steps:1. **Vector Decomposition:** - Decompose each vector into its horizontal (x) and vertical (y) components. Use trigonometric functions (sine and cosine) for this purpose: - For Vector \( \vec{A} \): \[ \begin{aligned} A_x &= -A \cos(\theta_A) = -3.30 \cos(35.0°) \\ A_y &= A \sin(\theta_A) = 3.30 \sin(35.0°) \end{aligned} \] - For Vector \( \vec{B} \): \[ \begin{aligned} B_x &= B \cos(\theta_B) = 7.40 \cos(12.0°) \\ B_y &= -B \sin(\theta_B) = -7.40 \sin(12.0°) \end{aligned} \] - For Vector \( \vec{C} \): \[ \begin{aligned} C_x &= C \cos(\theta_C) = 4.30 \cos(47.0°) \\ C_y &= -C \sin(\theta_C) = -4.30 \sin(47.0°) \end

A→ = 3.30 km @35 degree north of west B→ = 7.40 km@12 degree south of east C→ = 4.30 km@47 degree…

3. Consider the following three displacement vectors: Vector A has a magnitude of 3.30 km and a direction that makes an angle of 35.0° north of west, vector B has a magnitude of 7.40 km and a direction that makes an angle of 12.0° south of east, vector Ở has a magnitude of 4.30 km and a direction that makes an angle of 47.0° south of east. Determine the magnitude of the vector Z = C-B+ Ả km #60 ssfo ssf60 ƒ60 ssi ssf60"

3. Consider the following three displacement vectors: Vector A has a magnitude of 3.30 km and a direction that makes an angle of 35.0° north of west, vector B has a magnitude of 7.40 km and a direction that makes an angle of 12.0° south of east, vector Ở has a magnitude of 4.30 km and a direction that makes an angle of 47.0° south of east. Determine the magnitude of the vector Z = C-B+ Ả km #60 ssfo ssf60 ƒ60 ssi ssf60"

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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Scalars and Vectors

Scalars and Vectors are the classification of physical quantities that are used to denote various terms in physics.

Question

### Displacement Vectors Problem

Consider the following three displacement vectors:

1. **Vector \( \vec{A} \)** has a magnitude of 3.30 km and a direction that makes an angle of 35.0° north of west.
2. **Vector \( \vec{B} \)** has a magnitude of 7.40 km and a direction that makes an angle of 12.0° south of east.
3. **Vector \( \vec{C} \)** has a magnitude of 4.30 km and a direction that makes an angle of 47.0° south of east.

Determine the magnitude of the vector \( \vec{Z} = \vec{C} - \vec{B} + \vec{A} \).

**Answer:**
\[ \boxed{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{km} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \]

### Detailed Explanation of Steps:

1. **Vector Decomposition:**
- Decompose each vector into its horizontal (x) and vertical (y) components. Use trigonometric functions (sine and cosine) for this purpose:
- For Vector \( \vec{A} \):
\[
\begin{aligned}
A_x &= -A \cos(\theta_A) = -3.30 \cos(35.0°) \\
A_y &= A \sin(\theta_A) = 3.30 \sin(35.0°)
\end{aligned}
\]
- For Vector \( \vec{B} \):
\[
\begin{aligned}
B_x &= B \cos(\theta_B) = 7.40 \cos(12.0°) \\
B_y &= -B \sin(\theta_B) = -7.40 \sin(12.0°)
\end{aligned}
\]
- For Vector \( \vec{C} \):
\[
\begin{aligned}
C_x &= C \cos(\theta_C) = 4.30 \cos(47.0°) \\
C_y &= -C \sin(\theta_C) = -4.30 \sin(47.0°)
\end

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