Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Problem 1

Under what conditions is the magnitude of the vector sum \(A+B\) equal to the sum of the magnitudes of the two vectors?

Short Answer

Expert verified
The magnitude of the vector sum is equal to the sum of the magnitudes when the two vectors are in the same direction.

Step by step solution

01

Understand the Vector Magnitude Sum

The magnitude of the vector sum of two vectors, \(A+B\), is the length of the resultant vector formed by the vector addition of \(A\) and \(B\). The vector addition considers both the magnitudes (lengths) and the directions of the vectors.
02

Understand the Sum of Vector Magnitudes

The sum of the magnitudes of the two vectors, \(|A|+|B|\), means strictly adding up the lengths of the two vectors, regardless of their directions.
03

Compare and Establish the Condition

The magnitude of the vector sum is equal to the sum of the magnitudes (i.e., \(|A+B| = |A| + |B|\)) if and only if the two vectors are in the same direction. This is because if the vectors have different directions, some of the vector magnitude goes into changing the direction of the resultant vector, thereby reducing the overall magnitude.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

A flock of geese is attempting to migrate due south, but the wind is blowing from the west at \(5.1 \mathrm{m} / \mathrm{s}\). If the birds can fly at \(7.5 \mathrm{m} / \mathrm{s}\) relative to the air, what direction should they head?

Is a projectile's speed constant throughout its parabolic trajectory?

Consider two projectiles launched on level ground with the same speed, at angles \(45^{\circ} \pm \alpha .\) Show that the ratio of their flight times is \(\tan \left(\alpha+45^{\circ}\right)\)

The sum of two vectors, \(\vec{A}+\vec{B},\) is perpendicular to their difference, \(\vec{A}-\vec{B}\). How do the vectors' magnitudes compare?

A diver leaves a 3 -m board on a trajectory that takes her \(2.5 \mathrm{m}\) above the board and then into the water 2.8 m horizontally from the end of the board. At what speed and angle did she leave the board?
See all solutions

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Read Explanation

Magnetism

Read Explanation

Electromagnetism

Read Explanation

Space Physics

Read Explanation

Nuclear Physics

Read Explanation

Electrostatics

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.

Sign-up for free