/*! 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 49 (a) Find the scalar product of t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 49

(a) Find the scalar product of the vectors \(a \hat{\imath}+b \hat{\jmath}\) and \(b \hat{\imath}-a \hat{\jmath}\) where \(a\) and \(b\) are arbitrary constants. (b) What's the angle between the two vectors?

Short Answer

Expert verified
The scalar product of the vectors is 0 and the angle between them is 90掳 or \(\frac{\pi}{2}\) radians.

Step by step solution

01

Compute the dot product

Start with the formula for the dot product: \(A \cdot B = a_1b_1 + a_2b_2\). Here, \(a_1 = a, a_2 = b, b_1 = b, b_2 = -a\). So, replacing these values in the scalar product equation gives us: \(A \cdot B = a*b + b*(-a)\). Simplifying this expression we get \(A \cdot B = ab - ab = 0\).
02

Calculate the magnitudes of the vectors

The magnitude or length of a vector \(A = a_1 \hat{\imath} + a_2 \hat{\jmath}\) is given by \(\|A\| = \sqrt{a_1^2 + a_2^2}\). Using this, calculate the magnitude of vectors A and B. For Vector A: \(\|A\| = \sqrt{a^2 + b^2}\) and Vector B: \(\|B\| = \sqrt{b^2 + (-a)^2} = \sqrt{a^2 + b^2}\)
03

Finding the angle

We are now ready to find the angle \( \theta \) between the vectors. Use the formula \(\cos(\theta) = \frac{A \cdot B}{\|A\| \|B\|}\). Substituting the obtained values we get: \(\cos(\theta) = \frac{0}{(\sqrt{a^2 + b^2})(\sqrt{a^2 + b^2})} = 0\). The arccosine (inverse cosine) of zero is \(90^{\circ}\) or \(\frac{\pi}{2}\) radians in standard units hence that's the angle between the vectors.

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

A watt-second is a unit of what quantity? Relate it to a more standard SI unit.

A machine does work at a rate given by \(P=c t^{2},\) where \(c=18 \mathrm{W} / \mathrm{s}^{2}\) and \(t\) is time. Find the work done between \(t=10 \mathrm{s}\) and \(t=20 \mathrm{s}\).

Does your car's kinetic energy change if you drive at constant speed for 1 hour?

What's the kinetic energy of a \(2.4 \times 10^{5}-\mathrm{kg}\) airplane cruising at \(900 \mathrm{km} / \mathrm{h} ?\)

Show that the scalar product obeys the distributive law: \(\vec{A} \cdot(\vec{B}+\vec{C})=\vec{A} \cdot \vec{B}+\vec{A} \cdot \vec{C}\)
See all solutions

Recommended explanations on Physics Textbooks

Solid State Physics

Read Explanation

Famous Physicists

Read Explanation

Geometrical and Physical Optics

Read Explanation

Magnetism

Read Explanation

Turning Points in Physics

Read Explanation

Magnetism and Electromagnetic Induction

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.