/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 You are given two vectors \(\ove... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 40

You are given two vectors \(\overrightarrow{A}\) = \(-\)3.00\(\hat{\imath}\) \(+\) 6.00\(\hat{\jmath}\) and \(\overrightarrow{B}\) = 7.00\(\hat{\imath}\) \(+\) 2.00\(\hat{\jmath}\). Let counterclockwise angles be positive. (a) What angle does \(\overrightarrow{A}\) make with the \(+\)x-axis? (b) What angle does \(\overrightarrow{B}\) make with the \(+\)x-axis? (c) Vector \(\overrightarrow{C}\) is the sum of \(\overrightarrow{A}\) and \(\overrightarrow{B}\), so \(\overrightarrow{C}\) = \(\overrightarrow{A}\) \(+\) \(\overrightarrow{B}\). What angle does \(\overrightarrow{C}\) make with the \(+\)x-axis?

Short Answer

Expert verified
(a) 116.57掳 (b) 15.95掳 (c) 63.43掳

Step by step solution

01

Understand the Task

We need to find the angle each vector makes with the positive x-axis. Given vectors are \( \overrightarrow{A} = -3.00\hat{\imath} + 6.00\hat{\jmath} \) and \( \overrightarrow{B} = 7.00\hat{\imath} + 2.00\hat{\jmath} \). We compute the angle using the arctangent function for each vector and for \( \overrightarrow{C} \), which is the sum of \( \overrightarrow{A} \) and \( \overrightarrow{B} \).
02

Find Angle for \( \overrightarrow{A} \)

To find the angle \( \theta_{A} \) that \( \overrightarrow{A} \) makes with the x-axis, use the formula \( \theta = \arctan\left(\frac{y}{x}\right) \), where \( x = -3.00 \) and \( y = 6.00 \) are the components of \( \overrightarrow{A} \). Thus, \( \theta_{A} = \arctan\left(\frac{6.00}{-3.00}\right) = \arctan(-2) \). Carry out this calculation to find \( \theta_{A} \).
03

Calculate \( \theta_{A} \)

Calculate \( \theta_{A} = \arctan(-2) \). Since the arctangent of a negative number indicates the vector is in the second or fourth quadrants, you need to add 180 degrees for counterclockwise calculation: \( \theta_{A} = \arctan(-2) + 180^{\circ} \approx 116.57^{\circ} \).
04

Find Angle for \( \overrightarrow{B} \)

Use the same formula to find the angle \( \theta_{B} \) for vector \( \overrightarrow{B} \): \( \theta_{B} = \arctan\left(\frac{2.00}{7.00}\right) \). Calculate \( \theta_{B} \).
05

Calculate \( \theta_{B} \)

Calculate \( \theta_{B} = \arctan\left(\frac{2.00}{7.00}\right) \approx 15.95^{\circ} \). This angle is in the first quadrant.
06

Find Vector \( \overrightarrow{C} \)

Find vector \( \overrightarrow{C} \) by summing the components of \( \overrightarrow{A} \) and \( \overrightarrow{B} \): \( \overrightarrow{C} = (-3.00 + 7.00)\hat{\imath} + (6.00 + 2.00)\hat{\jmath} = 4.00\hat{\imath} + 8.00\hat{\jmath} \).
07

Find Angle for \( \overrightarrow{C} \)

Use the arctangent formula to find the angle \( \theta_{C} \): \( \theta_{C} = \arctan\left(\frac{8.00}{4.00}\right) \). Calculate \( \theta_{C} \).
08

Calculate \( \theta_{C} \)

Calculate \( \theta_{C} = \arctan\left(\frac{8.00}{4.00}\right) = \arctan(2) \approx 63.43^{\circ} \). This angle is also in the first quadrant.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Arctangent Function
When working with vectors, it鈥檚 essential to know how to find angles they make with specific axes. One useful tool here is the arctangent function, often written as \( \arctan \). This function helps us to find the angle of a right-angled triangle when the length of the opposite side and the length of the adjacent side are known. In vector mathematics, these sides come from vector components.
  • To calculate the angle \( \theta \) a vector makes with the x-axis, use the formula \( \theta = \arctan\left(\frac{y}{x}\right) \) where \( y \) and \( x \) are the vector's components along the y-axis and x-axis, respectively.
  • It鈥檚 essential to consider the sign of the components since it affects the direction and subsequently the quadrant in which the angle lies.
  • Remember that \( \arctan \) provides angles between \(-90^{\circ}\) and \(90^{\circ}\), but you may need adjustments if your vector is not in the first quadrant.
This math concept is crucial when solving vector problems as it provides an accurate way to calculate angles in coordinate systems.
Exploring Vector Addition
Vectors represent quantities that have both magnitude and direction. When dealing with more than one vector, vector addition becomes essential to determine the result of their combination, known as the resultant vector.

To add vectors:
  • Add their corresponding components along each axis. For example, if you have two vectors, \( \overrightarrow{A} \) and \( \overrightarrow{B} \), the resultant vector \( \overrightarrow{C} = \overrightarrow{A} + \overrightarrow{B} \).
  • This can be shown mathematically as \( \overrightarrow{C} = ( A_x + B_x )\hat{\imath} + ( A_y + B_y )\hat{\jmath} \), where \( A_x, A_y \) are the components of vector \( \overrightarrow{A} \) and \( B_x, B_y \) are the components of vector \( \overrightarrow{B} \).

Vector addition allows us to compute the total effect of multiple vectors acting at a point, which can widely apply in physics and engineering fields.
Breaking Down Vector Components
Vector components are fundamental in simplifying vector problems. A vector can be split into parts along the x-axis and y-axis, known as its components. These components make it easier to perform various calculations, like addition or finding angles.
  • The component along the x-axis is given by \( x = |\overrightarrow{V}| \cdot \cos(\theta) \), and along the y-axis, it鈥檚 \( y = |\overrightarrow{V}| \cdot \sin(\theta) \).
  • When components are known, it becomes straightforward to reconstruct the vector: the vector \( |\overrightarrow{V}| \) is the magnitude, and \( \theta \) is the angle with the positive x-axis.


In this context, recognizing and correctly using vector components facilitates the handling of more complex vector operations, acting as a building block in vector mathematics.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Two workers pull horizontally on a heavy box, but one pulls twice as hard as the other. The larger pull is directed at 21.0\(^{\circ}\) west of north, and the resultant of these two pulls is 460.0 N directly northward. Use vector components to find the magnitude of each of these pulls and the direction of the smaller pull.

Let \(\theta\) be the angle that the vector \(\overrightarrow{A}\) makes with the \(+$$x\)-axis, measured counterclockwise from that axis. Find angle \(\theta\) for a vector that has these components: (a) A\(_x\) = 2.00 m, A\(_y\) = \(-\)1.00 m; (b) A\(_x\) = 2.00 m, A\(_y\) = 1.00 m; (c) A\(_x\) = \(-\)2.00 m, A\(_y\) = 1.00 m; (d) A\(_x\) = -2.00 m, A\(_y\) = -1.00 m.

In each case, find the \(x\)- and \(y\)-components of vector \(\overrightarrow{A}\): (a) \(\overrightarrow{A}\) = 5.0\(\hat{\imath}\) \(-\) 6.3\(\hat{\jmath}\); (b) \(\overrightarrow{A}\) = 11.2\(\hat{\jmath}\) \(-\) 9.91\(\hat{\imath}\); (c) \(\overrightarrow{A}\) = \(-\)15.0\(\hat{\imath}\) \(+\) 22.4\(\hat{\jmath}\) ; (d) \(\overrightarrow{A}\) = 5.0\(\overrightarrow{B}\), where \(\overrightarrow{B}\) = 4\(\hat{\imath}\) \(+\) 6\(\hat{\jmath}\).

Express each approximation of \(\pi\) to six significant figures: (a) 22/7 and (b) 355/113. (c) Are these approximations accurate to that precision?

Four astronauts are in a spherical space station. (a) If, as is typical, each of them breathes about 500 cm\(^3\) of air with each breath, approximately what volume of air (in cubic meters) do these astronauts breathe in a year? (b) What would the diameter (in meters) of the space station have to be to contain all this air?
See all solutions

Recommended explanations on Physics Textbooks

Particle Model of Matter

Read Explanation

Fluids

Read Explanation

Thermodynamics

Read Explanation

Oscillations

Read Explanation

Work Energy and Power

Read Explanation

Translational Dynamics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.