Understanding the vector and the magnitude of a vector
When studying mathematics at high school, students have to deal with a topic concerning the magnitude of a vector. The term magnitude is commonly used not only in mathematics, but also in physics and is an integral part of the overall course of the high school education program. In terms of mathematics, the magnitude describes the size of particular mathematical object. This is a kind of property which is used when it comes to comparing that or another object with other objects and identify whether it is bigger or smaller than other objects, which are also of the similar type. In other words, the magnitude is used when one needs to rank the category of particular objects to which the given object (the one that is being compared) belongs. Many assignments on mathematics ask students to find the magnitude of a vector. Dealing with these assignments has a lot of methods, which depends on the data provided in the assignment and on the complexity level of the assignment. In this article, you will get familiar with the common ways to find the magnitude and learn some essentials which will help you understand this aspect of mathematics better.
A good starting point before you learn what is the magnitude would be figuring out the meaning and the common definition of the term «vector». The word «vector» may refer to various meanings, used in different aspects of mathematics, including geometry, and physics. Generally speaking, without referring to any specific area of science, a vector is considered to be a component of a vector space. When talking of a vector space, which is sometimes is also called a linear space, one should mean a number (a set) of certain objects of the same class. These objects are called vectors. They can be scaled (multiplied) by different numbers and also they can be added one to another. The numbers that the vectors can be multiplied by are called scalars, and are characterized by all kinds of real numbers that can be calculated by means of one real number. In case if you don’t know in terms of what area of mathematics you are dealing with that or another vector, the provided above definition is acceptable to apply.
When it comes to geometry, linear algebra as a branch of mathematics, and physics, the definition of a vector may be a little different. First of all, the vector used in geometry is called a Euclidean vector. Second of all, its properties and characteristics are also different. This kind of vector is used to describe physical quantities, both of which are considered to have a particular direction and magnitude. Euclidean vectors represent a vivid example of a vector space that was mentioned earlier in the article. This kind of vectors are expressed in so-called forces, which are specific physical quantities. Within the vector space, you can take two elements (forces) of the same type and add them one to another and add them to yield a third. Having said that, it is not necessarily that the elements of the vector space are visual objects in the form of arrows, although this is how they usually are represented. On the contrary, they can be abstract; elements of the vector space can be mathematical objects with certain characteristics and properties, which can be expressed in the form of arrows, but at the same time remain abstract.
The magnitude of a vector serves to represent the size of the objects the explanation of which is provided above. It helps compare various elements of the same class but with different characteristics between each other and find out which one is larger or smaller, etc.
The usage of a vector in different disciplines
Depending on the discipline where the term magnitude of a vector is used, it has different purposes and characterized differently. In terms of linear algebra, vector spaces are characterized in a different manner from how they are characterized in terms of geometry, for example. Their common and the main characteristic, when it comes to linear algebra, is the dimension. As a matter of fact, this characteristic is usually used in order to specify the particular number of separate directions located within the vector space that are independent one from another. Also, there may be extra structures within the vector space, which are expressed in the form of a so-called inner product or norm product. As a rule, spaces like these occur in such branch of mathematics as mathematical analysis, which is dedicated to the study of continuos changes and contains a number of different theories, including but not limited to analytic functions, integration, infinite series and others. Analytical problems occurring within the study of mathematical analysis require capability of identifying various phenomena, like whether a set of given elements of the vector space converges to a particular element.
If to discuss the vector space through the prism of history, it has to be mentioned that the first discussions concerning the subject of vector spaces took place in the seventeenth century. At that times, this issue has been researched within various aspects of mathematics, including analytic geometry, Euclidean vectors, matrices, the theory of systems and linear equations (the subject of study of linear algebra) and so on. In contemporary world, there is also a wide range of areas where the usage of vector spaces is a common occurrence. Among such areas, are not only mathematics and physics, but also science, engineering. In addition, since the beginning of computer era, vector spaces have been used in computer science also.
How to deal with finding the magnitude of a vector
Now that you are aware of essentials concerning vectors as a subject of study of mathematics and particularly geometry, you can proceed to learning how to deal with finding of the magnitude of a vector. Such parameter as vector is often faced in a wide range of geometry and physics problems. As a matter of fact, students find such assignments quite difficult to accomplish, because they require not only lots of theoretical knowledge, but also ability to think in an analytical manner. As for the magnitude, it can be explained as the length of the given vector. At the same time, the direction is considered to be the point at which the vector is directed. The methodology of measuring the magnitude of a vector consists of a number of steps that are described below. Read them attentively and follow when completing your assignment.
- The first step you will have to undertake will be to identify the components, which the given vector consists of. Each vector may appear to be express in a numerical manner in the Cartesian system of coordinates with elements located on the both fields of the coordinate system, the horizontal and vertical.
- Make up a pair of coordinates. There are two components that the given in the assignment vector has. These components are horizontal and vertical. For instance. If the given vector has a vertical component expressed in the number -3 and the horizontal component expressed in the number 7, you will have to make up the following pair of coordinates: <7; -3>.
- The next step that you will have to undertake is drawing a vector triangle. Once you have figured out the components of the given vector and you have drawn both horizontal and vertical component, you will notice that there is a right triangle in front of you. You may connect the points (the angles) of the triangle in order to be able to visualize it better, although it is not necessary.
- Keep in mind that the magnitude of a vector should be equal to the longest side of the triangle (its hypotenuse). Here, you will have to remember the Pythagorean theorem in order to find out the magnitude. Right down the theorem into your textbook.
- Now that you have the needed theorem written, you will have to rearrange it in order to calculate the magnitude of a vector. As you have already know the theorem looks like the following: AxA + BxB = CxC. In this equation, the letters A and B represent the two kinds of components of the triangle, the horizontal and the vertical one. As for the C, it represents the side with the biggest length of the triangle (the hypotenuse).
- Remember that you are going to find the hypotenuse because this is exactly the vector you are looking for. Therefore, you will have to solve the equation and find the C, which is the side with the biggest length..
- After the equation is solved, you will have to deal with the magnitude. Here, you will need to use the equation that you have already written, which is described in the previous paragraph.
- Once the equation is solved, look at the number and identify whether it is a whole number or not. In case if it is not a whole number, there is nothing to worry about, since the magnitude can be expressed through the decimals.
One more method to find the magnitude
The method provided above is good to use when you need to find the magnitude at the origin. On the other hand, however, there are other methods, depending on the information and requirements provided in the assignment. For example, if you need to find the magnitude from the origin, you will have to undertake the following steps. Read them carefully.
- First of all, you will need to identify the components that belong to the both points of the given in the assignment vector. Keep in mind that each vector can be expressed through the numbers in the Cartesian system of coordinates with two components, where the first component is horizontal and the second component is vertical.
- Remember that the components of the given vector have to be written in a pair. In case if the assignment provides you with a vector located far from the origin of the system of coordinates, it is necessary to indicate the components that belong to the both points of the given vector.
- Now you will need to apply the modified formula in order to calculate the magnitude of a vector. Since you have two points now that you are coping with, you will have to subtract the both components that belong to every point before you actually proceed to calculating the magnitude by solving the equation.
- The final step will be to find the magnitude. You will have to enter the numbers of the pairs of components and find out the magnitude. As we said while explaining the previous method of finding the magnitude, the answer to your equation doesn’t necessarily have to be the whole number, since the magnitude can be decimals.
These are the guidelines that will help you deal with your assignment requiring to find the magnitude. Following them will make it easier for you to deal with it. At the same time, remember about the opportunity to use professional assistance.
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