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