/*! 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 29 The angle between \(\vec{P}+\vec... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 29

The angle between \(\vec{P}+\vec{Q}\) and \(\vec{P}-\vec{Q}\) will be (A) \(90^{\circ}\) (B) Between \(0^{\circ}\) and \(180^{\circ}\) (C) \(180^{\circ}\) only (D) None of these

Short Answer

Expert verified
The correct option is (B) The angle between \(\vec{P}+\vec{Q}\) and \(\vec{P}-\vec{Q}\) is between \(0^{\circ}\) and \(180^{\circ}\).

Step by step solution

01

Calculating the Dot Product

The first step is to calculate the dot product of the vectors \(\vec{P}+\vec{Q}\) and \(\vec{P}-\vec{Q}\). The dot product is given by the formula \(\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos{\theta}\), where \(\theta\) is the angle between vectors Here, let's take \(\vec{A} = \vec{P} + \vec{Q}\) and \(\vec{B} = \vec{P} - \vec{Q}\).Compute the dot product: \((\vec{P} + \vec{Q}) \cdot (\vec{P} - \vec{Q}) = \vec{P} \cdot \vec{P} - \vec{Q} \cdot \vec{Q}\).
02

Observing the Result

The dot product \((\vec{P}+\vec{Q}) \cdot (\vec{P} - \vec{Q}) = \vec{P} \cdot \vec{P} - \vec{Q} \cdot \vec{Q}\) shows that this value is always a real number, as the dot product of a vector with itself gives the square of its magnitude which can never be a complex or imaginary number. Furthermore, if both vectors \(\vec{P}\) and \(\vec{Q}\) are non zero, this dot product can never be zero, therefore the cosine of the angle between \(\vec{P}+\vec{Q}\) and \(\vec{P}-\vec{Q}\) cannot be zero or undefined.
03

Concluding the Result

Based on observation in Step 2, it shows that the angle between \(\vec{P}+\vec{Q}\) and \(\vec{P}-\vec{Q}\) is neither \(90^{\circ}\) nor \(180^{\circ}\) as cos\(90^{\circ}\) = 0 and cos\(180^{\circ}\) = -1. Therefore, the angle must be between \(0^{\circ}\) and \(180^{\circ}\).

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!

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

The respective number of significant figure for the numbers \(23.023,0.0003\) and \(2.1 \times 10^{-3}\) are (A) \(4,4,2\) (B) \(5,1,2\) (C) \(5,1,5\) (D) \(5,5,2\)

The velocity of water waves may depend on their wavelength \(\lambda\), the density of water \(\rho\) and the acceleration due to gravity \(g\). The method of dimensions gives the relation between these quantities as (A) \(v^{2} \propto \lambda g^{-1} \rho^{-1}\) (B) \(v^{2} \propto g \lambda\) (C) \(v^{2} \propto \lambda g \rho\) (D) \(v^{2} \propto g^{-1} \lambda^{2}\)

The equation of state of some gases can be expressed as \(\left(P+\frac{a}{V^{2}}\right)(V-b)=R T .\) Here, \(P\) is the pressure, \(V\) the volume, \(T\) the absolute temperature, and \(a, b, R\) are constants. The dimensions of \(a\) are (A) \(\left[M L^{5} T^{-2}\right]\) (B) \(\left[M L^{-1} T^{-2}\right]\) (C) \(\left[M^{0} L^{3} T^{0}\right]\) (D) \(\left[M^{0} L^{6} T^{0}\right]\)

Find the dimensions of \(\frac{B^{2}}{\mu_{o}}\) (A) \(M L^{2} T^{-2}\) (B) \(M L^{-1} T^{-1}\) (C) \(M L^{-2} T^{-2}\) (D) \(M L^{-1} T^{-2}\)

If \(\vec{b}=3 \hat{i}+4 \hat{j}\) and \(\vec{a}=\hat{i}-\hat{j}\), the vector having the same magnitude as that of \(\vec{b}\) and parallel to \(\vec{a}\) is (A) \(\frac{5}{\sqrt{2}}(\hat{i}-\hat{j})\) (B) \(\frac{5}{\sqrt{2}}(\hat{i}+\hat{j})\) (C) \(5(\hat{i}-\hat{j})\) (D) \(5(\hat{i}+\hat{j})\)
See all solutions

Recommended explanations on Physics Textbooks

Force

Read Explanation

Thermodynamics

Read Explanation

Translational Dynamics

Read Explanation

Wave Optics

Read Explanation

Electric Charge Field and Potential

Read Explanation

Conservation of Energy and Momentum

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.