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  • Describe a plane vector, using correct notation.
  • Perform basic vector operations (scalar multiplication, addition, subtraction).
  • Express a vector in component form.
  • Explain the formula for the magnitude of a vector.
  • Express a vector in terms of unit vectors.
  • Give two examples of vector quantities.

When describing the movement of an airplane in flight, it is important to communicate two pieces of information: the direction in which the plane is traveling and the plane’s speed. When measuring a force, such as the thrust of the plane’s engines, it is important to describe not only the strength of that force, but also the direction in which it is applied. Some quantities, such as or force, are defined in terms of both size (also called magnitude ) and direction. A quantity that has magnitude and direction is called a vector    . In this text, we denote vectors by boldface letters, such as v .


A vector is a quantity that has both magnitude and direction.

Vector representation

A vector in a plane is represented by a directed line segment (an arrow). The endpoints of the segment are called the initial point    and the terminal point    of the vector. An arrow from the initial point to the terminal point indicates the direction of the vector. The length of the line segment represents its magnitude    . We use the notation v to denote the magnitude of the vector v . A vector with an initial point and terminal point that are the same is called the zero vector    , denoted 0 . The zero vector is the only vector without a direction, and by convention can be considered to have any direction convenient to the problem at hand.

Vectors with the same magnitude and direction are called equivalent vectors. We treat equivalent vectors as equal, even if they have different initial points. Thus, if v and w are equivalent, we write

v = w .


Vectors are said to be equivalent vectors    if they have the same magnitude and direction.

The arrows in [link] (b) are equivalent. Each arrow has the same length and direction. A closely related concept is the idea of parallel vectors. Two vectors are said to be parallel if they have the same or opposite directions. We explore this idea in more detail later in the chapter. A vector is defined by its magnitude and direction, regardless of where its initial point is located.

This figure has two images. The first is labeled “a” and has a line segment representing vector v. The line segment begins at the initial point and goes to the terminal point. There is an arrowhead at the terminal point. The second image is labeled “b” and is five vectors, each labeled v sub 1, v sub 2, v sub 3, v sub 4, v sub 5. They all are pointing in the same direction and have the same length.
(a) A vector is represented by a directed line segment from its initial point to its terminal point. (b) Vectors v 1 through v 5 are equivalent.

The use of boldface, lowercase letters to name vectors is a common representation in print, but there are alternative notations. When writing the name of a vector by hand, for example, it is easier to sketch an arrow over the variable than to simulate boldface type: v . When a vector has initial point P and terminal point Q , the notation P Q is useful because it indicates the direction and location of the vector.

Sketching vectors

Sketch a vector in the plane from initial point P ( 1 , 1 ) to terminal point Q ( 8 , 5 ) .

See [link] . Because the vector goes from point P to point Q , we name it P Q .

This figure is a graph of the first quadrant. There is a line segment beginning at the ordered pair (1, 1). Also, this point is labeled “P.” The line segment ends at the ordered pair (8, 5) and is labeled “Q.” The line segment is labeled “PQ.”
The vector with initial point ( 1 , 1 ) and terminal point ( 8 , 5 ) is named P Q .
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Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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