**Vector Addition Explanation**In this exercise, we will explore the addition of two vectors, \(\vec{A}\) and \(\vec{B}\), and determine the resultant vector \(\vec{A} + \vec{B}\).**Vector Details:**- **Vector \(\vec{A}\):** - Magnitude: 8.0 meters - Direction: Points east- **Vector \(\vec{B}\):** - Magnitude: 5.0 meters - Direction: Points 30° west of north**Objective:**Determine the resultant vector \(\vec{A} + \vec{B}\).To solve this, represent each vector in component form, apply the principles of vector addition, and calculate the resultant vector's magnitude and direction. ### Steps to Calculate the Resultant Vector:1. **Convert Each Vector into Components:** - Vector \(\vec{A}\) components (as it points east): - \(A_x = 8.0 \, \text{m}\) - \(A_y = 0\) - Vector \(\vec{B}\) components: - Using trigonometry, calculate each component based on the angle provided (30° west of north): - \(B_x = 5.0 \, \text{m} \times \sin(30^\circ)\) (west, which is negative in the x-direction) - \(B_y = 5.0 \, \text{m} \times \cos(30^\circ)\)2. **Add Components to Find the Resultant Vector:** - Resultant vector components: - \(R_x = A_x + B_x\) - \(R_y = A_y + B_y\)3. **Calculate the Magnitude and Direction:** - Magnitude: \(R = \sqrt{R_x^2 + R_y^2}\) - Direction: \(\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right)\)This exercise helps in understanding vector addition, breaking vectors into components, and using trigonometry to solve real-world physics problems.

Given data The magnitude of the vector A is 8 m. The magnitude of vector B is 5 m and points 30°…

Answered: Vector A has a magnitude of 8.0 m and…

Vector A has a magnitude of 8.0 m and points east, vector B has a magnitude of 5.0 m and points 30° West of North The resultant vector A + B is given by

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

**Vector Addition Explanation**

In this exercise, we will explore the addition of two vectors, \(\vec{A}\) and \(\vec{B}\), and determine the resultant vector \(\vec{A} + \vec{B}\).

**Vector Details:**

- **Vector \(\vec{A}\):**
- Magnitude: 8.0 meters
- Direction: Points east

- **Vector \(\vec{B}\):**
- Magnitude: 5.0 meters
- Direction: Points 30° west of north

**Objective:**
Determine the resultant vector \(\vec{A} + \vec{B}\).

To solve this, represent each vector in component form, apply the principles of vector addition, and calculate the resultant vector's magnitude and direction.

### Steps to Calculate the Resultant Vector:

1. **Convert Each Vector into Components:**
- Vector \(\vec{A}\) components (as it points east):
- \(A_x = 8.0 \, \text{m}\)
- \(A_y = 0\)

- Vector \(\vec{B}\) components:
- Using trigonometry, calculate each component based on the angle provided (30° west of north):
- \(B_x = 5.0 \, \text{m} \times \sin(30^\circ)\) (west, which is negative in the x-direction)
- \(B_y = 5.0 \, \text{m} \times \cos(30^\circ)\)

2. **Add Components to Find the Resultant Vector:**
- Resultant vector components:
- \(R_x = A_x + B_x\)
- \(R_y = A_y + B_y\)

3. **Calculate the Magnitude and Direction:**
- Magnitude: \(R = \sqrt{R_x^2 + R_y^2}\)
- Direction: \(\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right)\)

This exercise helps in understanding vector addition, breaking vectors into components, and using trigonometry to solve real-world physics problems.

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