/*! 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 28 Find a vector of magnitude 5 tha... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 28

Find a vector of magnitude 5 that is parallel to \(-12 \mathbf{i}+9 \mathbf{j}\)

Short Answer

Expert verified
-4\mathbf{i} + 3\mathbf{j}

Step by step solution

01

Find the magnitude of the given vector

To find a vector of magnitude 5 that is parallel to p{-12 \mathbf{i} + 9 \mathbf{j}}, start by determining the magnitude of the given vector using the formula for the magnitude of a vector \mathbf{v} = a\mathbf{i} + b\mathbf{j} which is \[ |\mathbf{v}| = \sqrt{a^2 + b^2} \]. So for the vector \mathbf{v} = -12\mathbf{i} + 9\mathbf{j}, the magnitude is \[ |\mathbf{v}| = \sqrt{(-12)^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15 \].
02

Determine the unit vector in the direction of the given vector

The unit vector \mathbf{u} in the direction of \mathbf{v} = -12\mathbf{i} + 9\mathbf{j} is found by dividing each component of the vector by its magnitude 15. This gives \[ \mathbf{u} = \frac{-12}{15} \mathbf{i} + \frac{9}{15} \mathbf{j} = -\frac{4}{5}\mathbf{i} + \frac{3}{5}\mathbf{j} \].
03

Scale the unit vector to the desired magnitude

To find a vector with magnitude 5 that is parallel to \mathbf{v}, multiply the unit vector by 5.This results in \[ 5\mathbf{u} = 5 \left(-\frac{4}{5} \mathbf{i} + \frac{3}{5} \mathbf{j} \right) = -4 \mathbf{i} + 3 \mathbf{j} \].

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 calculation
When working with vectors, one of the fundamental tasks is calculating their magnitude. The magnitude of a vector is essentially its length and can be found by using the Pythagorean theorem generalized for vectors. For a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \), the magnitude \( |\mathbf{v}| \) is calculated as \[ |\mathbf{v}| = \sqrt{a^2 + b^2} \].
In the given exercise, we start with the vector \( -12\mathbf{i} + 9\mathbf{j} \).
To find its magnitude, plug in the components into the formula: \[ |\mathbf{v}| = \sqrt{(-12)^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15 \].
This tells us that the original vector has a length of 15 units.
unit vector
A unit vector is a vector with a magnitude of 1. It points in the same direction as the original vector but has been normalized to a length of 1. To find the unit vector \( \mathbf{u} \) in the direction of \( \mathbf{v} = -12\mathbf{i} + 9\mathbf{j} \), you divide each component of the vector by its magnitude. This results in:
\[ \mathbf{u} = \frac{-12}{15} \mathbf{i} + \frac{9}{15} \mathbf{j} = -\frac{4}{5} \mathbf{i} + \frac{3}{5} \mathbf{j} \].
The unit vector has the same direction as the original but a magnitude of 1. This is crucial for scaling vectors, as it allows us to easily adjust the vector's length while maintaining its direction.
scaling vectors
After finding the unit vector, the next step is to scale it to the desired magnitude. In our exercise, we need a vector of magnitude 5 that is parallel to \( -12\mathbf{i} + 9\mathbf{j} \). This is done by multiplying the unit vector by the desired magnitude:
\[ 5\mathbf{u} = 5 \left(-\frac{4}{5} \mathbf{i} + \frac{3}{5} \mathbf{j}\right) = -4 \mathbf{i} + 3 \mathbf{j} \].
Now, we have a vector \( -4\mathbf{i} + 3\mathbf{j} \) that is 5 units long and points in the same direction as the original vector. This process of scaling vectors is very useful when you need vectors of a specific length for applications in physics, engineering, and computer science.

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

Solar Energy The amount of energy collected by a solar panel depends on the intensity of the sun's rays and the area of the panel. Let the vector I represent the intensity, in watts per square centimeter, having the direction of the sun's rays. Let the vector \(\mathbf{A}\) represent the area, in square centimeters, whose direction is the orientation of a solar panel. See the figure. The total number of watts collected by the panel is given by \(W=|\mathbf{I} \cdot \mathbf{A}|\) Suppose that \(\mathbf{I}=\langle-0.02,-0.01\rangle\) and \(\mathbf{A}=\langle 300,400\rangle\) (a) Find \(\|\mathbf{I}\|\) and \(\|\mathbf{A}\|,\) and interpret the meaning of each. (b) Compute \(W\) and interpret its meaning. (c) If the solar panel is to collect the maximum number of watts, what must be true about I and \(\mathbf{A}\) ?

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=3 \mathbf{i}-4 \mathbf{j}, \quad \mathbf{w}=9 \mathbf{i}-12 \mathbf{j} $$

A Boeing 787 Dreamliner maintains a constant airspeed of 550 miles per hour (mph) headed due north. The jet stream is \(100 \mathrm{mph}\) in the northeasterly direction. (a) Express the velocity \(\mathbf{v}_{\mathrm{a}}\) of the 787 relative to the air and the velocity \(\mathbf{v}_{\mathrm{w}}\) of the jet stream in terms of \(\mathbf{i}\) and \(\mathbf{j}\). (b) Find the velocity of the 787 relative to the ground. (c) Find the actual speed and direction of the 787 relative to the ground.

Find a vector of magnitude 15 that is parallel to \(4 \mathbf{i}-3 \mathbf{j}\)

Decompose \(\mathbf{v}\) into two vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\), and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\). $$ \mathbf{v}=3 \mathbf{i}+\mathbf{j}, \quad \mathbf{w}=-2 \mathbf{i}-\mathbf{j} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

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.