Search

Log in

Latest topics

Statistics

We have **30**registered users

The newest registered user is

**deigavr**

Our users have posted a total of

**191**messages in

**126**subjects

Who is online?

In total there are **2**users online :: 0 Registered, 0 Hidden and 2 Guests

None

Most users ever online was

**34**on Wed Nov 04, 2009 7:39 am

# SCALER AND VECTOR FULL INFORMATION

## SCALER AND VECTOR FULL INFORMATION

BASIC:::::::::::::::.......:::::::......

This is a basic, though hopefully fairly comprehensive, introduction to

working with vectors. Vectors manifest in a wide variety of ways, from

displacement, velocity and acceleration to forces and fields. This

article is devoted to the mathematics of vectors; their application in

specific situations will be addressed elsewhere.

In everyday conversation, when we discuss a quantity we are generally discussing a

which has only a magnitude. If we say that we drive 10 miles, we are

talking about the total distance we have traveled. Scalar variables

will be denoted, in this article, as an italicized variable, such as

A

about not just the magnitude but also the direction of the quantity.

When giving directions to a house, it isn't enough to say that it's 10

miles away, but the direction of those 10 miles must also be provided

for the information to be useful. Vector variables will be indicated

with a boldface variable, although it is common to see a vector denoted

with a small arrow above it.

Just as we don't say the other house is -10 miles away, the

magnitude of a vector is always a positive number, or rather the

absolute value of the "length" of the vector (although the quantity may

not be a length, it may be a velocity, acceleration, force, etc.) A

negative in front a vector doesn't indicate a change in the magnitude,

but rather in the direction of the vector. In the examples above, distance is the scalar quantity (10 miles) but

is the vector quantity (10 miles to the northeast). Similarly, speed is

a scalar quantity while velocity is a vector quantity.

A

vector representing a unit vector is usually also boldface, although it

will have a carat (

The

Vectors are generally oriented on a coordinate system, the most

popular of which is the two-dimensional Cartesian plane. The Cartesian

plane has a horizontal axis which is labeled x and a vertical axis

labeled y. Some advanced applications of vectors in physics require

using a three-dimensional space, in which the axes are x, y, and z.

This article will deal mostly with the two-dimensional system, though

the concepts can be expanded with some care to three dimensions without

too much trouble.

Vectors in multiple-dimension coordinate systems can be broken up into their

determine the magnitude of the components, you apply rules about

triangles that are learned in your math classes. Considering the angle

(the name of the Greek symbol for the angle in the drawing) between the

x-axis (or x-component) and the vector. If we look at the right

triangle that includes that angle, we see that

the direction of the components, but we're trying to find their

magnitude, so we strip away the directional information and perform

these scalar calculations to figure out the magnitude. Further

application of trigonometry can be used to find other relationships

(such as the tangent) relating between some of these quantities, but I

think that's enough for now.

For many years, the only mathematics that a student learns is scalar

mathematics. If you travel 5 miles north and 5 miles east, you've

traveled 10 miles. Adding scalar quantities ignores all information

about the directions.

Vectors are manipulated somewhat differently. The direction must always be taken into account when manipulating them.

When you add two vectors, it is as if you took the vectors and

placed them end to end, and created a new vector running from the

starting point to the end point, as demonstrated in the picture to the

right. If the vectors have the same direction, then this just means

adding the magnitudes, but if they have different directions, it can

become more complex.

You add vectors by breaking them into their components and then adding the components, as below:

The two x-components will result in the x-component of the new

variable, while the two y-components result in the y-component of the

new variable.

The order in which you add the vectors does not matter (as

demonstrated in the picture). In fact, several properties from scalar

addition hold for vector addition:

This is a basic, though hopefully fairly comprehensive, introduction to

working with vectors. Vectors manifest in a wide variety of ways, from

displacement, velocity and acceleration to forces and fields. This

article is devoted to the mathematics of vectors; their application in

specific situations will be addressed elsewhere.

**Vectors & Scalars**In everyday conversation, when we discuss a quantity we are generally discussing a

*scalar quantity*,which has only a magnitude. If we say that we drive 10 miles, we are

talking about the total distance we have traveled. Scalar variables

will be denoted, in this article, as an italicized variable, such as

*a*.A

*vector quantity*, or*vector*, provides informationabout not just the magnitude but also the direction of the quantity.

When giving directions to a house, it isn't enough to say that it's 10

miles away, but the direction of those 10 miles must also be provided

for the information to be useful. Vector variables will be indicated

with a boldface variable, although it is common to see a vector denoted

with a small arrow above it.

Just as we don't say the other house is -10 miles away, the

magnitude of a vector is always a positive number, or rather the

absolute value of the "length" of the vector (although the quantity may

not be a length, it may be a velocity, acceleration, force, etc.) A

negative in front a vector doesn't indicate a change in the magnitude,

but rather in the direction of the vector. In the examples above, distance is the scalar quantity (10 miles) but

*displacement*is the vector quantity (10 miles to the northeast). Similarly, speed is

a scalar quantity while velocity is a vector quantity.

A

*unit vector*is a vector that has a magnitude of one. Avector representing a unit vector is usually also boldface, although it

will have a carat (

**^**) above it to indicate the unit nature of the variable. The unit vector*, when written with a carat, is generally read as "x-hat" because the carat looks kind of like a hat on the variable.***x**The

*zero vector*, or*null vector*, is a vector with a magnitude of zero. It is written as**0**in this article.**Vector Components**Vectors are generally oriented on a coordinate system, the most

popular of which is the two-dimensional Cartesian plane. The Cartesian

plane has a horizontal axis which is labeled x and a vertical axis

labeled y. Some advanced applications of vectors in physics require

using a three-dimensional space, in which the axes are x, y, and z.

This article will deal mostly with the two-dimensional system, though

the concepts can be expanded with some care to three dimensions without

too much trouble.

Vectors in multiple-dimension coordinate systems can be broken up into their

*component vectors*. In the two-dimensional case, this results in a*x-component*and a*y-component*. The picture to the right is an example of a Force vector (**F**) broken into its components (**F**&_{x}**F**). When breaking a vector into its components, the vector is a sum of the components:_{y}ToF=F+_{x}F_{y}

determine the magnitude of the components, you apply rules about

triangles that are learned in your math classes. Considering the angle

*theta*(the name of the Greek symbol for the angle in the drawing) between the

x-axis (or x-component) and the vector. If we look at the right

triangle that includes that angle, we see that

**F**is the adjacent side,_{x}**F**is the opposite side, and_{y}**F**is the hypotenuse. From the rules for right triangles, we know then that:Note that the numbers here are the magnitudes of the vectors. We knowF/_{x}F= costhetaandF/_{y}F= sintheta

which gives usF=_{x}FcosthetaandF=_{y}Fsintheta

the direction of the components, but we're trying to find their

magnitude, so we strip away the directional information and perform

these scalar calculations to figure out the magnitude. Further

application of trigonometry can be used to find other relationships

(such as the tangent) relating between some of these quantities, but I

think that's enough for now.

For many years, the only mathematics that a student learns is scalar

mathematics. If you travel 5 miles north and 5 miles east, you've

traveled 10 miles. Adding scalar quantities ignores all information

about the directions.

Vectors are manipulated somewhat differently. The direction must always be taken into account when manipulating them.

**Adding Components**When you add two vectors, it is as if you took the vectors and

placed them end to end, and created a new vector running from the

starting point to the end point, as demonstrated in the picture to the

right. If the vectors have the same direction, then this just means

adding the magnitudes, but if they have different directions, it can

become more complex.

You add vectors by breaking them into their components and then adding the components, as below:

a+b=ca+_{x}a+_{y}b+_{x}b=_{y}

(a+_{x}b) + (_{x}a+_{y}b) =_{y}c+_{x}c_{y}

The two x-components will result in the x-component of the new

variable, while the two y-components result in the y-component of the

new variable.

**Properties of Vector Addition**The order in which you add the vectors does not matter (as

demonstrated in the picture). In fact, several properties from scalar

addition hold for vector addition:

Identity Property of Vector Additiona+0=aInverse Property of Vector Additiona+ -a=a-a=0Reflective Property of Vector Additiona=a

Commutative Property of Vector Additiona+b=b+aAssociative Property of Vector Addition

(a+b) +c=a+ (b+c)Transitive Property of Vector Addition

Ifa=bandc=b, thena=c

The simplest operation that can be performed on a vector is to multiply

it by a scalar. This scalar multiplication alters the magnitude of the

vector. In other word, it makes the vector longer or shorter.

When multiplying times a negative scalar, the resulting vector will point in the opposite direction.

Examples of scalar multiplication by 2 and -1 can be seen in the diagram...

Thescalar productof two vectors is a way to multiply them

together to obtain a scalar quantity. This is written as a

multiplication of the two vectors, with a dot in the middle

representing the multiplication. As such, it is often called thedot productof two vectors.

To calculate the dot product of two vectors, you consider the

angle between them, as shown in the diagram. In other words, if they

shared the same starting point, what would be the angle measurement (theta) between them. The dot product is defined as:a*b=abcostheta

In other words, you multiply the magnitudes of the two vectors, then multiply by the cosine of the angle separation. Thoughaandb

- the magnitudes of the two vectors - are always positive, cosine

varies so the values can be positive, negative, or zero. It should also

be noted that this operation is commutative, soa*b=b*a.

In cases when the vectors are perpendicular (ortheta= 90 degrees), costhetawill be zero. Therefore,the dot product of perpendicular vectors is always zero. When the vectors are parallel (ortheta= 0 degrees), costhetais 1, so the scalar product is just the product of the magnitudes.

These neat little facts can be used to prove that, if you know

the components, you can eliminate the need for theta entirely, with the

(two-dimensional) equation:a*b=a+_{x}b_{x}aThe_{y}b_{y}vector productis written in the formaxb, and is usually called thecross product

of two vectors. In this case, we are multiplying the vectors and

instead of getting a scalar quantity, we will get a vector quantity.

This is the trickiest of the vector computations we'll be dealing with,

as it iscommutative and involves the use of the dreadednotright-hand rule, which I will get to shortly.Calculating the Magnitude

Again, we consider two vectors drawn from the same point, with the anglethetabetween them (see picture to right). We always take the smallest angle, sotheta

will always be in a range from 0 to 180 and the result will, therefore,

never be negative. The magnitude of the resulting vector is determined

as follows:Ifc=axb, thenc=absintheta

When the vectors are parallel, sinthetawill be 0, sothe vector product of parallel (or antiparallel) vectors is always zero. Specifically, crossing a vector with itself will always yield a vector product of zero.Direction of the Vector

Now that we have the magnitude of the

vector product, we must determine what direction the resulting vector

will point. If you have two vectors, there is always a plane (a flat,

two-dimensional surface) which they rest in. No matter how they are

oriented, there's always one plane that includes them both. (This is a

basic law of Euclidean geometry.)

The vector product will be perpendicular to the plane created from

those two vectors. If you picture the plane as being flat on a table,

the question becomes will the resulting vector go up (our "out" of the

table, from our perspective) or down (or "into" the table, from our

perspective)?The Dreaded Right-Hand Rule

In order to figure this out, you must apply what is called theright-hand rule. When I studied physics in school, Idetested

the right-hand rule. Flat out hated it. Every time I used it, I had to

pull out the book to look up how it worked. Hopefully my description

will be a bit more intuitive than the one I was introduced to which, as

I read it now, still reads horribly.

If you haveaxb, as in the image to the right, you will place your right hand along the length ofbso that your fingers (except the thumb) can curve to point alonga. In other words, you are sort of trying to make the angletheta

between the palm and four fingers of your right hand. The thumb, in

this case, will be sticking straight up (or out of the screen, if you

try to do it up to the computer). Your knuckles will be roughly lined

up with the starting point of the two vectors. Precision isn't

essential, but I want you to get the idea since I don't have a picture

of this to provide.

If, however, you are consideringbxa, you will do the opposite. You will put your right hand alongaand point your fingers alongb.

If trying to do this on the computer screen, you will find it

impossible, so use your imagination. You will find that, in this case,

your imaginative thumb is pointing into the computer screen. That is

the direction of the resulting vector.

The right-hand rule shows the following relationship:axb= -bxa

Now that you have the means of finding the direction ofc=axb, you can also figure out the components ofc:c=_{x}a-_{y}b_{z}a_{z}b_{y}c=_{y}a-_{z}b_{x}a_{x}b_{z}

c=_{z}a-_{x}b_{y}a_{y}b_{x}

Notice that in the case whenaandbare entirely in the x-y plane (which is the easiest way to work with them), their z-components will be 0. Therefore,c&_{x}cwill equal zero. The only component of_{y}cwill be in the z-direction - out of or into the x-y plane - which is exactly what the right-hand rule showed us!Final Words

Don't be intimidated by vectors. When you're first

introduced to them, it can seem like they're overwhelming, but some

effort and attention to detail will result in quickly mastering the

concepts involved. At higher levels, vectors can get extremely complex to work with.

Entire courses in college, such as linear algebra, devote a great deal

of time to matrices (which I kindly avoided in this introduction),

vectors, andvector spaces. That level of detail is beyond the

scope of this article, but this should provide the foundations

necessary for most of the vector manipulation that is performed in the

physics classroom. If you are intending to study physics in greater

depth, you will be introduced to the more complex vector concepts as

you proceed through your education.

_________________

World In Hands

Similar topics

» Dream Theatre-Octavarium Full Album Zip

» [IMPLEMENTED] WinReducer Config Page: Full Automatic Installation

» FUNNY FULL FORMS

» Adding A Floating Menu!

» How to using "selenium rest web services" can any body provide sample script

» [IMPLEMENTED] WinReducer Config Page: Full Automatic Installation

» FUNNY FULL FORMS

» Adding A Floating Menu!

» How to using "selenium rest web services" can any body provide sample script

Page

**1**of**1****Permissions in this forum:**

**cannot**reply to topics in this forum

Mon Aug 01, 2011 11:44 pm by Guest

» Netbook Brands

Mon Aug 01, 2011 5:39 am by Guest

» backlinks checker backlink service

Sun Jul 31, 2011 8:56 am by Guest

» how to buy facebook fans f4

Sat Jul 30, 2011 2:34 pm by Guest

» Alle bijzondere dingen in de zaanstreek

Sat Jul 30, 2011 8:11 am by Guest

» HERE YOU CAN POST ALL WEBSITE LINKS...

Wed Aug 26, 2009 2:16 pm by onesimpletech

» cool web site for all must check it

Mon Jul 06, 2009 2:32 pm by Snopy Cobra

» poetry spirit

Mon Jul 06, 2009 2:31 pm by Snopy Cobra

» best to learn VC++

Sun May 10, 2009 3:22 pm by Guest