/*! 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 9 Find the magnitude and direction... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 9

Find the magnitude and direction angle of the given vector. $$u=(5,-1)$$

Short Answer

Expert verified
Magnitude is \( \sqrt{26} \) and direction angle is approximately \( 348.69^\circ \).

Step by step solution

01

Find the Magnitude of the Vector

The magnitude of a vector \( u = (a, b) \) is calculated using the formula: \( \|u\| = \sqrt{a^2 + b^2} \). For vector \( u = (5, -1) \), substitute \( a = 5 \) and \( b = -1 \) into the formula.\[ \|u\| = \sqrt{5^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26} \]This gives us a magnitude of \( \sqrt{26} \).
02

Determine the Direction Angle

The direction angle \( \theta \) of a vector \( u = (a, b) \) is found using the formula: \( \theta = \tan^{-1}(\frac{b}{a}) \). Here \( a = 5 \) and \( b = -1 \). Calculate:\[ \theta = \tan^{-1}\left(\frac{-1}{5}\right) = \tan^{-1}(-0.2) \]This angle \( \theta \) is the direction angle measured counter-clockwise from the positive x-axis. Use a calculator to find \( \theta \approx -11.31^\circ \). Since the angle is negative, to find a positive angle, add \( 360^ irc \): \( 360 - 11.31 = 348.69^\circ \).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnitude of a Vector
The magnitude of a vector is essentially its length. It tells you how long the vector is when measured from its start point to its endpoint. This can be particularly important in various applications, such as physics, engineering, and computer graphics. To find the magnitude, we use the formula:
  • \( \|u\| = \sqrt{a^2 + b^2} \)
Where \( u \) is the vector \((a, b)\). Simply substitute the given values into the formula. For the vector \( u = (5, -1) \), you calculate as follows:
  • \( \|u\| = \sqrt{5^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26} \)
This means the magnitude of the vector \( u \) is \( \sqrt{26} \), which is a little over 5, as \( \sqrt{25}=5 \). This understanding of magnitude helps visualize the vector's size on a graph.
Direction Angle
The direction angle of a vector gives us the orientation of the vector relative to the positive x-axis. It's crucial because it allows us to see in which direction the vector is pointing. The direction angle \( \theta \) can be found using the formula:
  • \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \)
This approach uses the inverse tangent function, which helps you find an angle whose tangent is \( \frac{b}{a} \). For our vector \( u = (5, -1) \):
  • \( \theta = \tan^{-1}\left(\frac{-1}{5}\right) = \tan^{-1}(-0.2) \)
Upon finding \( \theta \), you get approximately \( -11.31^\circ \). This value is negative because the vector points in the "below the x-axis" direction. To convert it to a positive angle often used in navigation and graphics, add \( 360^\circ \):
  • \( 360^\circ - 11.31^\circ = 348.69^\circ \)
Trigonometric Functions
To understand vectors better, trigonometric functions are very useful. These functions relate the angles and sides of triangles and are not only pivotal in geometry, but also in analyzing vectors.
  • Tangent (\( \tan \)): Used to relate the opposite side and adjacent side of a right triangle. In vectors, \( \tan 胃 = \frac{b}{a} \). This relationship is key in finding the direction angle of a vector.
  • Inverse Tangent (\( \tan^{-1} \)): Also known as \( \arctan \), is used to find the angle \( 胃 \) when you know the ratio of the sides.
When we apply \( \tan^{-1} \) in the context of a vector, we are solving for the angle that would give us the slope or inclination of the vector. This is why knowing trigonometric functions enhances our capability to interpret and manipulate vectors in various fields of study.

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

Determine whether each statement is true or false. All cardioids are lima莽ons, but not all lima莽ons are cardioids.

Suppose that \(\mathbf{u}=\langle 2000,5000\rangle\) represents the number of units of battery \(A\) and \(B\), respectively, produced by a company and \(\mathbf{v}=\langle 8.40,6.50\rangle\) represents the price (in dollars) of a 10 -pack of battery \(A\) and \(B\), respectively. Compute and interpret \(\mathbf{u} \cdot \mathbf{v}\).

Find the indicated dot product with a calculator. $$\langle 23,-350\rangle \cdot\langle 45,202\rangle$$

Determine whether each pair of vectors is orthogonal. $$\langle-7,3\rangle \text { and }\left\langle\frac{1}{7},-\frac{1}{3}\right\rangle$$

A truck that weighs 40,000 pounds is parked on a hill in San Francisco with a slant of \(10^{\circ}\) from the horizontal. How much force will keep it from rolling down the hill?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

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.