<< Chapter < Page Chapter >> Page >
In this figure a vector C with a negative slope is drawn from the origin. Then from the head of the vector C another vector A with positive slope is drawn and then another vector B with negative slope from the head of the vector A is drawn. From the tail of the vector C a vector R of magnitude of fifty point zero meters and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector B. The vector R is known as the resultant vector.

Here, we see that when the same vectors are added in a different order, the result is the same. This characteristic is true in every case and is an important characteristic of vectors. Vector addition is commutative    . Vectors can be added in any order.

A + B = B + A . size 12{"A+B=B+A"} {}

(This is true for the addition of ordinary numbers as well—you get the same result whether you add 2 + 3 size 12{"2+3"} {} or 3 + 2 size 12{"3+2"} {} , for example).

Vector subtraction

Vector subtraction is a straightforward extension of vector addition. To define subtraction (say we want to subtract B size 12{B} {} from A size 12{A} {} , written A B size 12{ "A" "-B"} {} , we must first define what we mean by subtraction. The negative of a vector B is defined to be –B ; that is, graphically the negative of any vector has the same magnitude but the opposite direction , as shown in [link] . In other words, B size 12{B} {} has the same length as –B size 12{"-" "B"} {} , but points in the opposite direction. Essentially, we just flip the vector so it points in the opposite direction.

Two vectors are shown. One of the vectors is labeled as vector   in north east direction. The other vector is of the same magnitude and is in the opposite direction to that of vector B. This vector is denoted as negative B.
The negative of a vector is just another vector of the same magnitude but pointing in the opposite direction. So B size 12{B} {} is the negative of –B size 12{ ital "-B"} {} ; it has the same length but opposite direction.

The subtraction of vector B from vector A is then simply defined to be the addition of –B to A . Note that vector subtraction is the addition of a negative vector. The order of subtraction does not affect the results.

A – B = A +  ( –B ) . size 12{ bold "A – B = A + " \( bold "–B" \) } {}

This is analogous to the subtraction of scalars (where, for example, 5 – 2 = 5 +  ( –2 ) size 12{"5 – 2 = 5 + " \( "–2" \) } {} ). Again, the result is independent of the order in which the subtraction is made. When vectors are subtracted graphically, the techniques outlined above are used, as the following example illustrates.

Subtracting vectors graphically: a woman sailing a boat

A woman sailing a boat at night is following directions to a dock. The instructions read to first sail 27.5 m in a direction 66.0º size 12{"66" "." 0º} {} north of east from her current location, and then travel 30.0 m in a direction 112º size 12{"112"º} {} north of east (or 22.0º size 12{"22" "." 0º} {} west of north). If the woman makes a mistake and travels in the opposite direction for the second leg of the trip, where will she end up? Compare this location with the location of the dock.

A vector of magnitude twenty seven point five meters is shown. It is inclined to the horizontal at an angle of sixty six degrees. Another vector of magnitude thirty point zero meters is shown. It is inclined to the horizontal at an angle of one hundred and twelve degrees.

Strategy

We can represent the first leg of the trip with a vector A , and the second leg of the trip with a vector B size 12{B} {} . The dock is located at a location A + B . If the woman mistakenly travels in the opposite direction for the second leg of the journey, she will travel a distance B (30.0 m) in the direction 180º 112º = 68º south of east. We represent this as –B , as shown below. The vector –B has the same magnitude as B but is in the opposite direction. Thus, she will end up at a location A + ( –B ) , or A B .

A vector labeled negative B is inclined at an angle of sixty-eight degrees below a horizontal line. A dotted line in the reverse direction inclined at one hundred and twelve degrees above the horizontal line is also shown.

We will perform vector addition to compare the location of the dock, B size 12{ ital "A ""+ "B} {} , with the location at which the woman mistakenly arrives, A +  ( –B ) size 12{ bold "A + " \( bold "–B" \) } {} .

Solution

(1) To determine the location at which the woman arrives by accident, draw vectors A size 12{A} {} and –B .

(2) Place the vectors head to tail.

(3) Draw the resultant vector R size 12{R} {} .

(4) Use a ruler and protractor to measure the magnitude and direction of R size 12{R} {} .

Vectors A and negative B are connected in head to tail method. Vector A is inclined with horizontal with positive slope and vector negative B with a negative slope. The resultant of these two vectors is shown as a vector R from tail of A to the head of negative B. The length of the resultant is twenty three point zero meters and has a negative slope of seven point five degrees.

In this case, R = 23 . 0 m size 12{R"=23" "." "0 m"} {} and θ = 7 . size 12{θ=7 "." "5° south of east"} {} south of east.

(5) To determine the location of the dock, we repeat this method to add vectors A size 12{A} {} and B size 12{B} {} . We obtain the resultant vector R ' size 12{R'} {} :

A vector A inclined at sixty six degrees with horizontal is shown. From the head of this vector another vector B is started. Vector B is inclined at one hundred and twelve degrees with the horizontal. Another vector labeled as R prime from the tail of vector A to the head of vector B is drawn. The length of this vector is fifty two point nine meters and its inclination with the horizontal is shown as ninety point one degrees. Vector R prime is equal to the sum of vectors A and B.

In this case R  = 52.9 m size 12{R" = 52" "." "9 m"} {} and θ = 90.1º size 12{θ="90" "." "1° north of east "} {}  north of east.

We can see that the woman will end up a significant distance from the dock if she travels in the opposite direction for the second leg of the trip.

Discussion

Because subtraction of a vector is the same as addition of a vector with the opposite direction, the graphical method of subtracting vectors works the same as for addition.

Questions & Answers

memory of development brain of the human psychologist
SERAJ Reply
?
lord
hlo
Ananya
Haw are you?
Shilan
hi
zge
hello
Emm
usef
hi
Gil
Haw are you?
Shilan
Aha ok
Shilan
hi
Daniella
waasup
Isaiah
hello
Sara
hello
androi
Haw are you
Shilan
im ok you
Daniella
Me to
Shilan
Thanks
Shilan
What's going?
Shilan
working
Daniella
how did psychology begin?
Valerie Reply
of psychologys commencement, the traces can be seen in the work of Aristotle, where he talk about soul and body, likewise work in durrant, de anima, all these were somewhere supporting dualism, in which soul could exist separately from body
amaan
but if you talk about the moder psychology, Gustav fechner, is credited with performing scientific experiments, basis of his experiments in psychology with his studies perception.
amaan
does psychology deal with love?
Mohammed Reply
Maybe, i think
edem
I definitely would say yes
Clara
how so
Isaiah
*triarchic
Meredith
there are so many different reasons why you can fall in love with someone, many of them develope subconsciously -> psychology
Clara
love messes with the brain, a lot, ergo I believe that Psychology does indeed deal with love
what is synapse
Katie Reply
In the central nervous system, a synapse is a small gap at the end of a neuron that allows a signal to pass from one neuron to the next. synapse are found where nerve cells connect with other nerve cells
Najeem
a synapse the connection is where a neuron cell connects to another neuron cell.
Shaun
good
Jobe
what is psychology
Jobe
can you do auto book auto
Mariah Reply
WHT u mean?
usef
yes
MD
heyy, may i join the conversation please?
edem Reply
who is the father of psychology
Richy Reply
aristatil
Ramadevi
and please, how would you guys, describe the study of psychology at college ?
edem
psychologist student?
Aspen
i mean not yet but am about to start college so wanna know how is it(college in general and psychology course) please
edem
Psychology is the study of mind and behaviour. So if you will take psychology as a subject so you will get to know how your everything (physical, mental, social, spiritual aspects) effects your behaviour
sakina
With this brief knowledge you can help people to cope up with their problems and only you can guide them correctly
sakina
And if you go for further specialisations you can study hypnosis, face reading, body language etc
sakina
Thanks a lot🙏🏾 And ik some of the stuffs u said but i am also going to write thesis, right ?
edem
ok no prob, thanks a lot🙏🏾✨
edem
cerebellum
Khan
hae everyone, hope you are well this evning my question is what is the difference between drive and motivation
Michael
good question
Rainee
drive is more like an impulse or urge and i think they both go together (drive and motivation) even if there is a slight difference
edem
@ Michael Drive is delivered to be innate without the use of an external stimuli, motivation normally evolves an outside stimuli which may include praise, appreciate, or reward.
Reginald
*believed...sorry for typo
Reginald
@Reginald, can't the motivation come from the inner self?
edem
Good question, please give an example.
Reginald
can we say desire of success for example
edem
Wilhelm Wundt is the father of psychology
ipau
Wilhem Wundt thank you for the road that you opened.
Qwanta
You mean who is the father of having a great educated argumentative guess? nothing is more wrong than this question. The question is you should ask yourselfs is, how sure are you abour their scientific studying? one's percieved assimilated approach to judging another person and saying they are
Roger
the biggest problem with scientific research and data is that ya you could get the same result 1000 times then it could go the other way 1000 times, but we would never know that and we did, we would still say ya but the proof is there. The only thing science proves is that humanity has
Roger
no facts about human behavior in the scientific context, but more in the trial and error.. sorry to tell you, but so far no one has proven Father of anything, thats up to you and i, judgement is bias, science is good enough lazy
Roger
cognitive development is the growing and development of the brain.
Jessy Reply
Anyone knows about Techno-fascism?
Hussein Reply
Ecofascism is a theoretical political model in which an authoritarian government would require individuals to sacrifice their own interests to the "organic whole of nature". The term is also used as a rhetorical pejorative to undermine the environmental movement.
ipau
what's the big difference between prejudice and discrimination?
Danice Reply
A prejudiced person may not act on their attitude.  Therefore, someone can be prejudiced towards a certain group but not discriminate against them.  Also, prejudice includes all three components of an attitude (affective, behavioral and cognitive), whereas discrimination just involves behavior
Nancy Lee
hi
basher
hello
Rahul
what is all about cognitive development?
Kamohelo
cognitive development is the growing and development of the brain
Jessy
how do you control a variable when using spss whilst running a pearsons correlation analysis?
Jessie Reply
it dependa on your study. according to what you want to say and explain your result
Pouran
why does it say her and she
Jayla Reply
stages of cognitive development
brivia Reply
sensory preoperatinal concrete formal
Rajendra
what is psychology
Chethani Reply
the study of insecurities and the effect on the host .
Sera
Psychology is the scientific study of behavior & mental processes
Angela
psychology is science about learning human behaviour
Zhamshid
behaviorosm
Khan
In thinking about the case of Candace described earlier, do you think that Candace benefitted or suffered as a result of consistently being passed on to the next grade?
Nene Reply
An helicopter is flying over new York with a horizontal component of velocity of 14.6m/s-1 and a vertical component of -8.62 m/s-1, calculate, (a), the magnitude of the total velocity of the helicopter. (b), the angle of the total velocity.
Eseka Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, 2d kinematics. OpenStax CNX. Sep 04, 2015 Download for free at http://legacy.cnx.org/content/col11879/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the '2d kinematics' conversation and receive update notifications?

Ask