/*! 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 55 For \(\mathbf{v}=10 \mathbf{i}-2... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 55

For \(\mathbf{v}=10 \mathbf{i}-24 \mathbf{j},\) find \(\|\mathbf{v}\|\)

Short Answer

Expert verified
\(\big\| \textbf{v} \big\| = 26\)

Step by step solution

01

Identify the Components of the Vector

The given vector is \(\textbf{v} = 10\textbf{i} - 24\textbf{j}\). Here, the component along \(\textbf{i}\) (x-component) is 10 and the component along \(\textbf{j}\) (y-component) is -24.
02

Apply the Magnitude Formula

The formula to find the magnitude of a vector \(\textbf{v} \) with components \(a \) and \(b\) is \(\big\| \textbf{v} \big\| = \sqrt{a^2 + b^2}\). For our vector, \(a = 10\) and \(b = -24\).
03

Compute the Squares of the Components

Calculate the square of each component: \(10^2 = 100\) and \((-24)^2 = 576\).
04

Sum the Squares

Add the results from the previous step: \(100 + 576 = 676\).
05

Take the Square Root

Find the square root of the sum: \(\big\| \textbf{v} \big\| = \sqrt{676} = 26\).

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.

Vector Components
Vectors are quantities that have both magnitude and direction. In the context of a coordinate system, vectors can be broken down into components along the axes. For instance, if you have a vector \(\textbf{v} = 10\textbf{i} - 24\textbf{j}\), the components of this vector are:
  • 10 along the \(\textbf{i}\) (or x) axis
  • -24 along the \(\textbf{j}\) (or y) axis
. These components tell us how far the vector extends in each direction. Components are crucial because they allow us to analyze and work with vectors more easily. For example, to find a vector's total length, we can use its components in a formula.
Magnitude Formula
Once we know the components of a vector, we can calculate its magnitude, which tells us how long the vector is. The magnitude formula for a vector \( \textbf{v} = a\textbf{i} + b\textbf{j} \) is: \(\big\| \textbf{v} \big\| = \sqrt{a^2 + b^2}\). Here \(a\) and \(b\) are the components of the vector. Let's use our original vector \( \textbf{v} = 10\textbf{i} - 24\textbf{j} \). We identify the components as:
  • a = 10
  • b = -24
By plugging these values into the magnitude formula, we will find the vector's length.
Square Root Calculation
To complete the magnitude calculation, we need to perform a few steps involving square roots. First, we square each component: \(10^2 = 100\) and \((-24)^2 = 576\). Then, we add these squares together: \(\text{Sum} = 100 + 576 = 676\). Finally, we take the square root of this sum to get the magnitude: \(\big\| \textbf{v} \big\| = \sqrt{676} = 26\). The square root operation is crucial here because it converts the result back from a squared form to its original scale. Understanding the steps of squaring, summing, and then taking the square root helps us accurately determine the vector magnitude.

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

Use the fact that the orbit of a planet about the Sun is an ellipse, with the Sun at one focus. The aphelion of a planet is its greatest distance from the Sun, and the perihelion is its shortest distance. The mean distance of a planet from the Sun is the length of the semimajor axis of the elliptical orbit. A planet orbits a star in an elliptical orbit with the star located at one focus. The perihelion of the planet is 5 million miles. The eccentricity \(e\) of a conic section is \(e=\frac{c}{a} .\) If the eccentricity of the orbit is \(0.75,\) find the aphelion of the planet.

The arch of a bridge is a semiellipse with a horizontal major axis. The span is 30 feet, and the top of the arch is 10 feet above the major axis. The roadway is horizontal and is 2 feet above the top of the arch. Find the vertical distance from the roadway to the arch at 5 -foot intervals along the roadway.

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the horizontal asymptote for the graph of $$ f(x)=4 e^{x+1}-5 $$

Find the difference quotient of \(f(x)=2 x^{2}-7 x\) as \(h \rightarrow 0\).

Analyze each equation. \(\frac{(x-3)^{2}}{4}-\frac{y^{2}}{25}=1\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.