### Vector Addition and Direction CalculationIn this problem, we are given two vectors and need to determine the direction of their sum.#### Given Vectors:- Vector \( \mathbf{A} = 1.00\hat{i} - 2.00\hat{j} \)- Vector \( \mathbf{B} = 3.00\hat{i} + 4.00\hat{j} \)#### Task:What is the direction of vector \( \mathbf{C} = \mathbf{A} + \mathbf{B} \)?#### Options:- \( \circ \) 56.3° counterclockwise from the positive x-axis- \( \circ \) 26.6° counterclockwise from the positive x-axis- \( \circ \) 30.0° counterclockwise from the positive x-axis- \( \circ \) 60.0° counterclockwise from the positive x-axis- \( \circ \) 33.7° counterclockwise from the positive x-axis### Calculating the Sum of Vectors:To find vector \( \mathbf{C} \), we add corresponding components of vectors \( \mathbf{A} \) and \( \mathbf{B} \).\[ \mathbf{C} = \mathbf{A} + \mathbf{B} \]\[ \mathbf{C} = (1.00\hat{i} - 2.00\hat{j}) + (3.00\hat{i} + 4.00\hat{j}) \]\[ \mathbf{C} = (1.00 + 3.00)\hat{i} + (-2.00 + 4.00)\hat{j} \]\[ \mathbf{C} = 4.00\hat{i} + 2.00\hat{j} \]Now, we must find the direction of vector \( \mathbf{C} \). The direction θ is given by:\[ \theta = \tan^{-1} \left( \frac{C_y}{C_x} \right) \]Where \( C_y \) and \( C_x \) are the y and x components of vector \( \mathbf{C} \), respectively.\[ \theta = \tan^{-1} \left( \frac{2.00}{4.00} \right)

Answered: Image /qna-images/answer/445e1776-ff52-43c0-8717-dd4a5bf5d948.jpg

Vector A= 1.008 -2.00P, vector B =3.008 + 4.009. what is the direction of vector 56.3 counterclockwise from the positive x axis 26.6° counterclockwise from the positive x axis O 30.0° counterclockwise from the positive x axis 60.0° counterclockwise from the positive x axis O 33.7° counterclockwise from the positive x axis

Vector A= 1.008 -2.00P, vector B =3.008 + 4.009. what is the direction of vector 56.3 counterclockwise from the positive x axis 26.6° counterclockwise from the positive x axis O 30.0° counterclockwise from the positive x axis 60.0° counterclockwise from the positive x axis O 33.7° counterclockwise from the positive x axis

Physics for Scientists and Engineers: Foundations and Connections

1st Edition

ISBN:9781133939146

Author:Katz, Debora M.

Publisher:Katz, Debora M.

Chapter3: Vectors

Section: Chapter Questions

Problem 65PQ

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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 and Direction Calculation

In this problem, we are given two vectors and need to determine the direction of their sum.

#### Given Vectors:
- Vector \( \mathbf{A} = 1.00\hat{i} - 2.00\hat{j} \)
- Vector \( \mathbf{B} = 3.00\hat{i} + 4.00\hat{j} \)

#### Task:
What is the direction of vector \( \mathbf{C} = \mathbf{A} + \mathbf{B} \)?

#### Options:
- \( \circ \) 56.3° counterclockwise from the positive x-axis
- \( \circ \) 26.6° counterclockwise from the positive x-axis
- \( \circ \) 30.0° counterclockwise from the positive x-axis
- \( \circ \) 60.0° counterclockwise from the positive x-axis
- \( \circ \) 33.7° counterclockwise from the positive x-axis

### Calculating the Sum of Vectors:
To find vector \( \mathbf{C} \), we add corresponding components of vectors \( \mathbf{A} \) and \( \mathbf{B} \).

\[ \mathbf{C} = \mathbf{A} + \mathbf{B} \]
\[ \mathbf{C} = (1.00\hat{i} - 2.00\hat{j}) + (3.00\hat{i} + 4.00\hat{j}) \]
\[ \mathbf{C} = (1.00 + 3.00)\hat{i} + (-2.00 + 4.00)\hat{j} \]
\[ \mathbf{C} = 4.00\hat{i} + 2.00\hat{j} \]

Now, we must find the direction of vector \( \mathbf{C} \). The direction θ is given by:

\[ \theta = \tan^{-1} \left( \frac{C_y}{C_x} \right) \]

Where \( C_y \) and \( C_x \) are the y and x components of vector \( \mathbf{C} \), respectively.

\[ \theta = \tan^{-1} \left( \frac{2.00}{4.00} \right)

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