Vector Mathematics A Basic But Comprehensive Introduction

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 and Scalars A vector quantity, or vector, provides information about not just the magnitude but also the direction of the quantity. When giving directions to a house, it isnt enough to say that its 10 miles away, but the direction of those 10 miles must also be provided for the information to be useful. Variables that are vectors will be indicated with a boldface variable, although it is common to see vectors denoted with small arrows above the variable. Just as we dont 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 doesnt 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. A vector 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 x, when written with a carat, is generally read as x-hat because the carat looks kind of like a hat on the variable. 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. When breaking a vector into its components, the vector is a sum of the components: F Fx Fy thetaFxFyF Fx / F cos theta and Fy / F sin thetawhich gives usFx F cos theta and Fy F sin theta Note that the numbers here are the magnitudes of the vectors. We know the direction of the components, but were 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 thats 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, youve 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. 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 cax ay bx by ( ax bx) ( ay by) cx cy 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. 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 aCommutative Property of Vector Additiona b b aAssociative Property of Vector Addition(a b) c a (b c)Transitive Property of Vector AdditionIf a b and c b, then a 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 words, it makes the vector longer or shorter. When multiplying times a negative scalar, the resulting vector will point in the opposite direction. The scalar product of 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 the dot product of two vectors. To calculate the dot product of two vectors, you consider the angle between them. 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 ab cos theta ababba In cases when the vectors are perpendicular (or theta 90 degrees), cos theta will be zero. Therefore, the dot product of perpendicular vectors is always zero. When the vectors are parallel (or theta 0 degrees), cos theta is 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 ax bx ay by The vector product is written in the form a x b, and is usually called the cross 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 well be dealing with, as it is not commutative and involves the use of the dreaded right-hand rule, which I will get to shortly. Calculating the Magnitude Again, we consider two vectors drawn from the same point, with the angle theta between them. We always take the smallest angle, so theta 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: If c a x b, then c ab sin theta The vector product of parallel (or antiparallel) vectors is always zero Direction of the Vector 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 if 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 the right-hand rule. When I studied physics in school, I detested the right-hand rule. 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. If you have a x b you will place your right hand along the length of b so that your fingers (except the thumb) can curve to point along a. In other words, you are sort of trying to make the angle theta 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 isnt essential, but I want you to get the idea since I dont have a picture of this to provide. If, however, you are considering b x a, you will do the opposite. You will put your right hand along a and point your fingers along b. 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: a x b - b x a cabc cx ay bz - az bycy az bx - ax bzcz ax by - ay bx abcxcyc Final Words 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, and vector 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.

Pneumonia the Silent Killer - 895 Words

Running Head: PNEUMONIA: THE SILENT KILLER Exploring Pneumonia: A Silent Killer Abstract This paper explores Pneumonia and the respiratory disease process associated with bacterial and viral pathogens most commonly located in the lung. The paper examines the process, symptoms and treatments most commonly viewed in patient cases of Pneumonia. My goal is to educate the reader and to warn of the Exploring Pneumonia: A Silent Killer Pneumonia is an infection of the lungs. Many different organisms can cause it, including bacteria, viruses, and fungi. Pneumonia can range from mild to severe, and can even be deadly. The severity depends on the type of organism causing pneumonia, as well as your age and underlying health.†¦show more content†¦High-dose amoxicillin or amoxicillin-clavulanate. Many people can be treated at home with antibiotics. 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First aid impotance Free Essays

Here are eight reasons you should take Child CPR and First Aid: 1 . There are differences between Infant CPR and Child CPR, including: hand placement; the technique for compressions; how deep to push; the technique to give a breath. 2. We will write a custom essay sample on First aid impotance or any similar topic only for you Order Now Practice matters! We know from science that the retention of CPR skills is at best 6 months without review, and a change in skills complicates the issue. The American Heart Association recommends everyone should take a CPR class every two years; but f you only took Infant CPR, you haven’t yet learned what to do on a child older than one. . Your toddler is mobile. This developmental fact changes a lot! Injuries are now more common. And since a toddler is still gaining a sense of balance and his head is still heavy, the majority of falls in a toddler impact the head and face. Would you know what to do? 4. First Aid is used on a weekly, if not daily, basis for most toddlers! Think of everything your toddler gets into these days. Theyre more ndependent, exploring their world, and testing limits. We’ll discuss how to make their environment safer. Prevention is key. 5. Choking is more common. Young children, ages 1-5 are at a higher risk for choking since they are now eating new foods, and are also more likely to try to run while eating or not properly chew their food. The choke-saving skills are different for children than infants, so again that hands-on practice is important! 6. A chance to share experiences. In almost every Child CPR and First Aid class, there are a few people who have already been to the ER with their toddler or called their pediatrician with concerns over an injury. Learn from other parents! 7. This class is for ALL caregivers. Anyone who cares for your toddler needs to have this knowledge. Nannies, grandparents, parents. 8. You’ll have the skills to save a life. I saved my daughter’s life. As an American Heart Association instructor, as the Program Manager for CPR and Safety at Isis, and most importantly as a parent, I want you to have these skills, too. How to cite First aid impotance, Papers

Julius Caesar Essays (514 words) - Iulii, Julius Caesar,

Julius Caesar The Life of Julius Caesar GAIUS JULIUS CAESAR Julius Caesar was born on July 12 or July 13 100 BC. Since this was such a long time ago historians are not really sure on the exact date but the 12 and 13 are what must people think due what they know. People also know that Julius's father was also named Gaius Julius Caesar. Another thing people know of Caesar threw historians is that he had a brother in law who was related to him threw a marrige. In 84 BC Caesar married Cornelia, who was the daughter of Marius's old partner Lucius Cornelius. After Lucius ordering the couple to be divorced, Julius escaped harmed by the help of his mother and other relatives. Of course the two did not get divoced, because they loved each other so much. Caesar was then sent to collect a fleet from the Roman ally Nicomedes IV of Bithynia and was honored for conspicuous bravery at the siege of Mytilene. After awill he left Rome for studies in Rhodes but was captured on route by pirates. After obtaining ransom he recruited troops and captured the pirates then had them executed . His studies were then halted by the breakout of war with MITHRADATES VI of Ponus, against whom he hastily gathered a force. In Spain, after a quaestorship, Caesar earned great popularity among the Transpadane Gauls by supporting their agitation for Roman citizenship. He then married Pompey the granddaughter of Sulla and relative of POMPEY THE GREAT, evidence shows that may be Caesar supported important military assignments for Pompey in 67 and 66. He again came into great popularity then into debt by supporting splendid games. He probably also cooperated with Marcus Licinius in an attempt to annex Egypt. At one time Caesar was with political powers and many honors. He was appointed dictator. Then as time went on he was elected dictator for ten years. Then, finally, he became dictator for life. He was also elected consul, appointedperfect of morals, awarded tribunician sacrosanctity, and honored by having his face on coins and having a temple built for him. He also brought about many reforms so of which would are that he introduced the Julian Calendar, and enlarged the senate. At the same time he revised the tax structure, and maded it possably for non-Italians to become citizenes. Caesar was also an accomplished orator and writer. His two surviving works, On the Gallic War and On the Civil War, introduced the genre of personal war commentaries, Subtle propaganda for Caesaar, they are also lucid narratives that hold the reader. After writing this paper I learned many simalarities between the world we live in and the world Julius Caesar lived in. For example, in both then and now war is killing many people. The thing that brings war is greed over land, and may be even hatred. Another thing I learned was that we have a form of government consisting of a senete and may be the Romans influenced this. And even way back then they had writers such as Julius Caesar. And finally one of the things I learned was that they had a type of coin which presented the faces of famous men, in this case Gaius Julius Caesa.

Without one, half of our country would be in prison for crimes

An attorney is a privilege given to us by our government. Without one, half of our country would be in prison for crimes that they didn't commit. But when do you need a lawyer? And how do you go about finding one? Most people have no idea how to answer these questions, but everyone who lives in this country should know these answers. Why should anyone have this privilege and not know how to use it? I will discuss the steps needed to find and hire an attorney. First of all, a lawyer is needed in many situations. Some are easily apparent while others are not that obvious. The easiest to recognize is when you are charged with a crime, whether it be a felony or misdemeanor. A civil suit, usually for personal injury, is another time when an attorney is needed. In order to end a marriage, a lawyer is needed to get a divorce. After you've had a car accident and you insurance company refuses to pay, you may need to seek a lawyer's help. Before you purchase real estate, an attorney needs to review documents to determine the validity of the purchase. A person beginning a business or company needs to consult a lawyer in order to get all of the legal documents signed. Also, a person wanting to write a will or to plan their estate must have it legally bound by a lawyer. Big corporations and organizations also must have an attorney to prosper. An attorney's work is plentiful with all of these different specialities around. After you have determined that you need an attorney, you must find a lawyer that fits your legal needs. Many lawyers specialize in certain fields, while others have a wide range of abilities. But where do you look to find a lawyer? Many places are obvious. They are ordinary things found around the house. First of all, a phone book will give you a list of lawyers around that particular area. Another great way to find an attorney is in your local library. It houses a book,

Writing Contests Are Good Practice

Writing Contests Are Good Practice Writing Contests Are Good Practice Writing Contests Are Good Practice By Guest Author This is a guest post by Mary Hodges If you want to write for Daily Writing Tips check the guidelines here. There are some writers who wouldn’t dream of entering a writing competition. They consider such things beneath their dignity. I’ve even come across someone who refused to take part in competitions because â€Å"someone has to lose†. My response to this would be â€Å"someone has to win, and it might be you. But if you don’t enter, you certainly won’t win.† My own feeling is that writing for a competition is helpful for these reasons: You have to write to a word limit. You have a deadline to meet. You often have to write on a given topic. Even if your work is not among the winning entries you have a finished story article or poem that you can adapt for another market Writing competitions seem to be a growth area. A Google search for â€Å"writing competitions† came up with over a million hits! Prizemagic includes some interesting details about Michael Shenton and his book Stiff Competition based on his experience of entering competitions. The poetrykit site includes a link to story and other competitions and a warning to beware of scams. There are numerous print magazines with titles like Competitors’ Companion but these include only a few writing contests. They are mainly devoted to general consumer competitions where the entrants need to provide proof of purchase and concoct advertising slogans for particular products. Some points to remember when entering competitions: Always read the rules for entry carefully. State age, sex, ethnicity, place of birth or residence, and whether or not your work has been published. – I’ve seen all these used to restrict who is eligible to enter. Look carefully at the deadline. There’s no point in sending your work off too soon and making it unavailable for other uses any longer than necessary. Look for contests that give a clear date by which the winners will be notified. Once that date has passed, you know you can submit your work elsewhere or enter it in another competition, Note the prizes and the entry fees – if any. Remember, a bigger prize might attract more entries; you might have a better chance of winning in a smaller local comp. albeit with a smaller reward. Contests you can enter by email are to be preferred every time over those that require hard copy entries. Some ask you to send an S.A.E for an entry form, requiring wo lots of unnecessary postage. Look carefully at what the organizers say they will do with the non-winners. I’ve come across one contest where the entrants granted the organizers â€Å"a worldwide, irrevocable, perpetual license to feature any or all of the submissions in any of its publications†. Do you really want to give your work away when you might submit it to another publication or enter it for another contest? Keep a copy of your entry and a record of where you sent it. I should come clean and say I haven’t won lots of writing competitions. I seem to get to the runner-up stage quite often, but don’t make the big time. Of course prizes in writing competitions are not always in the form of cash. You might win a holiday, a place on a writing course in some exotic location or as I did many years ago you might win a lavatory seat! I’m not kidding. This was part of a prize for writing a poem about Convent Garden in London. Prizes were items from the various shops in Covent Garden including one that specialized in wooden loo seats with the owner’s initials on the lid! But when all’s said and done it’s not the winning that counts, it’s the taking part. Mary Hodges has published poems in The Oldie, Freelance Market News and Quantum Leap magazine. She edits the Garstang Arts Centre Newsletter and is Press Officer for her local Women’s Institute. She also writes plays, one of which was produced for the Women’s Institute Drama Festival. Although computer literate and interested in the web, she has not yet set up her own website or blog. Two sites she’s found particularly useful are PrizeMagic and The Poetry Kit. Want to improve your English in five minutes a day? Get a subscription and start receiving our writing tips and exercises daily! Keep learning! Browse the Freelance Writing category, check our popular posts, or choose a related post below:"Because Of" and "Due To" Comma Before ButA Yes-and-No Answer About Hyphenating Phrases

Take a position as to why kids join gangs Essay

Take a position as to why kids join gangs - Essay Example Some of them will still join the gangs even after being warned and having enough exposure to the appropriate advice against such behaviours. It therefore follows that there are some very pressing challenges that these children who join the gangs face, under the blanket of identity crisis during adolescence. The most potent of all factors leading to the stubborn insistence on social vices by adolescents and youths are internal factors that the developing personality tries to come to terms with. At such a time, these children require the appropriate conditioning that will encourage them to make the correct decision for instance in joining helpful friendships. In this paper, there are certain factors that appear to have the most determining forces that these children face leading to joining the wrong groups. Academic research shows that while some children find difficulties in cultivating the appropriate friendship patterns due to the environment that they are brought up in, there are chances that they would still behave the same if the environment was changed. Likewise, there are chances that environment would not affect yet another section of the youths’ population. The important thing is the realization that various factors contribute to marked differences in the life of the children. Kids join gangs for a number of reasons, among which are for protection, to gain of sense of belonging, or to get into illegal activities. Firstly, according to the Juvenile Justice Bulletin (1), children who are brought up in a neighbourhood where the environment for the cultivation of a positive social integration is poor always find the wrong footing into social life. The Bulletin finds strong links in the presence of gangs in such neighbourhoods to the lack of the appropriate positioning and empowerment for making social decisions in the youths. Poor neighbourhoods present challenging development factors for