/*! 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 50 Find a unit vector in the direct... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 50

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. $$\mathbf{v}=\langle-2,2\rangle$$

Short Answer

Expert verified
The unit vector in the direction of the vector \(\mathbf{v} = < -2, 2 >\) is \(\mathbf{u} = < -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} >\).

Step by step solution

01

Finding the Magnitude of the given vector

We'll begin by finding the magnitude of the given vector \(\mathbf{v} = <-2, 2 >\). The magnitude or norm of a vector \(\mathbf{v} = \) can be found using the formula \(\|\mathbf{v}\| = \sqrt{a^2 + b^2}\). Here, \(a = -2\) and \(b = 2\). So, the magnitude would be \(\sqrt{(-2)^2+(2)^2} = \sqrt{4+4} = \sqrt{8} = 2\sqrt{2}\).
02

Finding the Unit Vector

The next step is to find the unit vector in the same direction as the given vector \(\mathbf{v}\). We can achieve this by dividing each component of the vector by its magnitude. That is, the unit vector \(\mathbf{u}\) is given by \(\mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|}\). Thus, \(\mathbf{u}=<\frac{-2}{2\sqrt{2}},\frac{2}{2\sqrt{2}}>= <-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}>\).
03

Verifying the magnitude of the Unit Vector

The final step is to verify that the magnitude of the unit vector is 1. To do this, we must find the magnitude of vector \(\mathbf{u}\) using the formula given in Step 1. Therefore, the norm of \(\mathbf{u}\) can be calculated as \(\|\mathbf{u}\| = \sqrt{(-1/\sqrt{2})^2 + (1/\sqrt{2})^2}=\sqrt{1/2 + 1/2}= \sqrt{1}=1\). This verifies that the obtained vector is indeed a unit vector.

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

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.$$\mathbf{v}=\langle-1,1\rangle$$.

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.$$\mathbf{v}=\langle-24,-7\rangle$$

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$\theta=\frac{3 \pi}{4}$$

Find the magnitude and direction angle of the vector v. $$\mathbf{v}=12 \mathbf{i}+15 \mathbf{j}$$

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left[3\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)\right]^{4}$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.