10.8 Vectors (Page 5/22)

Page 5 / 22

Writing a vector in terms of i And j

Given a vector $v$ with initial point $P = (2, −6)$ and terminal point $Q = (−6, 6),$ write the vector in terms of $i$ and $j .$

Begin by writing the general form of the vector. Then replace the coordinates with the given values.

\begin{array}{l} v = (x_{2} - x_{1}) i + (y_{2} - y_{1}) j \\ = (- 6 - 2) i + (6 - (- 6)) j \\ = - 8 i + 12 j \end{array}

Got questions? Get instant answers now!

Writing a vector in terms of i And j Using initial and terminal points

Given initial point $P_{1} = (- 1, 3)$ and terminal point $P_{2} = (2, 7),$ write the vector $v$ in terms of $i$ and $j .$

Begin by writing the general form of the vector. Then replace the coordinates with the given values.

\begin{array}{l} v = (x_{2} - x_{1}) i + (y_{2} - y_{1}) j \\ v = (2 - (- 1)) i + (7 - 3) j \\ = 3 i + 4 j \end{array}

Got questions? Get instant answers now!

try it feature

Write the vector $u$ with initial point $P = (- 1, 6)$ and terminal point $Q = (7, - 5)$ in terms of $i$ and $j .$

$u = 8 i - 11 j$

Got questions? Get instant answers now!

Performing operations on vectors in terms of i And j

When vectors are written in terms of $i$ and $j,$ we can carry out addition, subtraction, and scalar multiplication by performing operations on corresponding components.

a general note label

Adding and subtracting vectors in rectangular coordinates

Given v = a i + b j and u = c i + d j , then

\begin{matrix} v + u = (a + c) i + (b + d) j \\ v - u = (a - c) i + (b - d) j \end{matrix}

Finding the sum of the vectors

Find the sum of $v_{1} = 2 i - 3 j$ and $v_{2} = 4 i + 5 j .$

According to the formula, we have

\begin{array}{l} v_{1} + v_{2} = (2 + 4) i + (- 3 + 5) j \\ = 6 i + 2 j \end{array}

Got questions? Get instant answers now!

Calculating the component form of a vector: direction

We have seen how to draw vectors according to their initial and terminal points and how to find the position vector. We have also examined notation for vectors drawn specifically in the Cartesian coordinate plane using $i and j .$ For any of these vectors, we can calculate the magnitude. Now, we want to combine the key points, and look further at the ideas of magnitude and direction.

Calculating direction follows the same straightforward process we used for polar coordinates. We find the direction of the vector by finding the angle to the horizontal. We do this by using the basic trigonometric identities, but with $| v |$ replacing $r .$

a general note label

Vector components in terms of magnitude and direction

Given a position vector $v = ⟨ x, y ⟩$ and a direction angle $θ,$

\begin{array}{l} \cos θ = \frac{x}{| v |} & and \begin{matrix}  \end{matrix} & \sin θ = \frac{y}{| v |} \\ x = | v | \cos θ \begin{matrix}  \end{matrix} & y = | v | \sin θ \end{array}

Thus, $v = x i + y j = | v | \cos θ i + | v | \sin θ j,$ and magnitude is expressed as $| v | = \sqrt{x^{2} + y^{2}} .$

Writing a vector in terms of magnitude and direction

Write a vector with length 7 at an angle of 135° to the positive x -axis in terms of magnitude and direction.

Using the conversion formulas $x = | v | \cos θ i$ and $y = | v | \sin θ j,$ we find that

\begin{array}{l} x = 7 \cos (135°) i \\ = - \frac{7 \sqrt{2}}{2} \\ y = 7 \sin (135°) j \\ = \frac{7 \sqrt{2}}{2} \end{array}

This vector can be written as $v = 7 \cos (135°) i + 7 \sin (135°) j$ or simplified as

v = - \frac{7 \sqrt{2}}{2} i + \frac{7 \sqrt{2}}{2} j

Got questions? Get instant answers now!

trey it feature

A vector travels from the origin to the point $(3, 5) .$ Write the vector in terms of magnitude and direction.

$v = \sqrt{34} \cos (59°) i + \sqrt{34} \sin (59°) j$

Magnitude = $\sqrt{34}$

$θ = \tan^{- 1} (\frac{5}{3}) = 59.04°$

Got questions? Get instant answers now!

Finding the dot product of two vectors

As we discussed earlier in the section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the dot product and the cross product . We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses.

Questions & Answers

how does Neisseria cause meningitis

Nyibol Reply

what is microbiologist

Muhammad Reply

what is errata

Muhammad

is the branch of biology that deals with the study of microorganisms.

Ntefuni Reply

What is microbiology

Mercy Reply

studies of microbes

Louisiaste

when we takee the specimen which lumbar,spin,

Ziyad Reply

How bacteria create energy to survive?

Muhamad Reply

Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments

_Adnan

But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?

Muhamad

they make spores

Louisiaste

what is sporadic nd endemic, epidemic

Aminu Reply

the significance of food webs for disease transmission

Abreham

food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.

Mark

explain assimilatory nitrate reduction

Esinniobiwa Reply

Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).

Elkana

This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products

Elkana

Examples of thermophilic organisms

Shu Reply

Give Examples of thermophilic organisms

Shu

advantages of normal Flora to the host

Micheal Reply

Prevent foreign microbes to the host

Abubakar

they provide healthier benefits to their hosts

ayesha

They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!

Mark

what is cell

faisal Reply

cell is the smallest unit of life

Fauziya

cell is the smallest unit of life

Akanni

Innocent

cell is the structural and functional unit of life

Hasan

is the fundamental units of Life

Musa

what are emergency diseases

Micheal Reply

There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️

_Adnan

define infection ,prevention and control

Innocent

I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life

Lubega

Heyy Lubega hussein where are u from?

_Adnan

en français

Adama

which site have a normal flora

ESTHER Reply

Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract

Safaa

skin

Asiina

skin,Oral,Nasal,GIt

Sadik

How can Commensal can Bacteria change into pathogen?

Sadik

How can Commensal Bacteria change into pathogen?

Sadik

all

Tesfaye

by fussion

Asiina

what are the advantages of normal Flora to the host

Micheal

what are the ways of control and prevention of nosocomial infection in the hospital

Micheal

what is inflammation

Shelly Reply

part of a tissue or an organ being wounded or bruised.

Wilfred

what term is used to name and classify microorganisms?

Micheal Reply

Binomial nomenclature

adeolu

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 11

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask

	11 Key concepts By OpenStax Read Online Course
	9 Calculating the component form of a vector: direction By OpenStax Read Online Course
	7 Microbiology Unit 3 By Madison Christian Start Test
	Financial Markets By Keyaira Braxton Start Exam
	10 AP 10 Muscle Tissue Essay Quiz By OpenStax Start Flashcards
©flickr: Jonathan	Power By Megan Earhart Start Quiz
	8 Neuroanatomy 08 The Vestibular System By Stephen Voron Start Quiz
	Social Dances 1 By Marion Cabalfin Start Test
	25 Toxicology Dr. Gustafson/Hamar quiz By Brooke Delaney Start Exam
	2 SCEA Java Architect By JavaChamp Team Start Exam
	43 Biology 43 Animal Reproduction Development MCQ By OpenStax Start Quiz
	2 Sociology 02 Sociological Research MCQ By OpenStax Start Quiz