/*! 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 56 Write an expression for a unit v... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 56

Write an expression for a unit vector at \(45^{\circ}\) clockwise from the \(x\) -axis.

Short Answer

Expert verified
The expression for the unit vector at \(45^{\circ}\) clockwise from the x-axis is \(\vec{v} = (\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\).

Step by step solution

01

Understanding the Concept of Unit Vector

A unit vector has a magnitude of 1. It is often used to represent directions. In a two-dimensional space with x and y axis, a unit vector can be expressed in terms of its x and y components. Let's denote the required unit vector as \(\vec{v}\).
02

Concept of Rotation

A positive rotation by an angle \(\theta\) counterclockwise from the positive x-axis or equivalently, a negative rotation by the same angle clockwise would result in a new vector whose x-component is \(\cos(\theta)\) and y-component is \(\sin(\theta)\).
03

Apply the Rotation

The given rotation is \(45^{\circ}\) clockwise which is equivalent to \(-45^{\circ}\) counterclockwise. Hence, the unit vector \(\vec{v}\) can be written as \( \vec{v} = (\cos(-45^{\circ}), \sin(-45^{\circ}))\).
04

Evaluate the Trigonometric Functions

\(\cos(-45^{\circ})\) = \(\cos(45^{\circ})\) = \(\frac{\sqrt{2}}{2}\) and \(\sin(-45^{\circ})\) = \(-\sin(45^{\circ})\) = \(-\frac{\sqrt{2}}{2}\)
05

Write the Answer in Vector Form

Replace the trigonometric function values in \(\vec{v}\) to get the unit vector \(\vec{v} = (\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\)

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 carpenter tosses a shingle horizontally off an 8.8 -m-high roof at \(11 \mathrm{m} / \mathrm{s} .\) (a) How long does it take the shingle to reach the ground? (b) How far does it move horizontally?

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?

A biologist looking through a microscope sees a bacterium at \(\vec{r}_{1}=2.2 \hat{\imath}+3.7 \hat{\jmath}-1.2 k \mu \mathrm{m} .\) After \(6.2 \mathrm{s},\) it's located at \(\vec{r}_{2}=4.6 \hat{\imath}+1.9 \hat{k} \mu \mathrm{m} .\) Find (a) its average velocity, expressed in unit vectors, and (b) its average speed.

Estimate the acceleration of the Moon, which completes a nearly circular orbit of 384.4 Mm radius in 27 days.

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

Recommended explanations on Physics Textbooks

Waves Physics

Read Explanation

Scientific Method Physics

Read Explanation

Nuclear Physics

Read Explanation

Magnetism

Read Explanation

Astrophysics

Read Explanation

Modern Physics

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.