Some physical and geometric quantities, called scalars, can be fully defined by specifying their magnitude in suitable units of measure. Thus, mass can be expressed in grams, temperature in degrees on some scale, and time in seconds. Scalars can be represented graphically by points on some numerical scale such as a clock or thermometer. There also are quantities, called vectors, that require the specification of direction as well as magnitude. Velocity, force , and displacement are examples of vectors. A vector quantity can be represented graphically by a directed line segment, symbolized by an arrow pointing in the direction of the vector quantity, with the length of the segment representing the magnitude of the vector.
A prototype of a vector is a directed line segment A B see Figure 1 that can be thought to represent the displacement of a particle from its initial position A to a new position B. To distinguish vectors from scalars it is customary to denote vectors by boldface letters. Thus the vector A B in Figure 1 can be denoted by a and its length or magnitude by a.
In many problems the location of the initial point of a vector is immaterial, so that two vectors are regarded as equal if they have the same length and the same direction. This construction of the sum, c , of a and b yields the same result as the parallelogram law in which the resultant c is given by the diagonal A C of the parallelogram constructed on vectors A B and A D as sides. Also, it is easy to show that the associative law. If s is a scalar, s a or a s is defined to be a vector whose length is s a and whose direction is that of a when s is positive and opposite to that of a if s is negative.
Thus, a and - a are vectors equal in magnitude but opposite in direction. The foregoing definitions and the well-known properties of scalar numbers represented by s and t show that. Inasmuch as the laws 1 , 2 , and 3 are identical with those encountered in ordinary algebra, it is quite proper to use familiar algebraic rules to solve systems of linear equations containing vectors. This fact makes it possible to deduce by purely algebraic means many theorems of synthetic Euclidean geometry that require complicated geometric constructions.
The multiplication of vectors leads to two types of products, the dot product and the cross product.
What are Vectors? (Easily Explained with 17+ Examples!)
The associative, commutative, and distributive laws of elementary algebra are valid for the dot multiplication of vectors. Also, since rotation from b to a is opposite to that from a to b ,. Since empirical laws of physics do not depend on special or accidental choices of reference frames selected to represent physical relations and geometric configurations, vector analysis forms an ideal tool for the study of the physical universe.
The introduction of a special reference frame or coordinate system establishes a correspondence between vectors and sets of numbers representing the components of vectors in that frame, and it induces definite rules of operation on these sets of numbers that follow from the rules for operations on the line segments. If some particular set of three noncollinear vectors termed base vectors is selected, then any vector A can be expressed uniquely as the diagonal of the parallelepiped whose edges are the components of A in the directions of the base vectors.
In common use is a set of three mutually orthogonal unit vectors i. In this system the expression takes the form. When two vectors A 1 and A 2 are represented as. Also, the dot product can be written. Such rephrasing suggests a generalization of the concept of a vector to spaces of dimensionality higher than three. For example, the state of a gas generally depends on the pressure p , volume v , temperature T , and time t. A quadruple of numbers p , v , T , t cannot be represented by a point in a three-dimensional reference frame.
Vectors in Physics
But since geometric visualization plays no role in algebraic calculations, the figurative language of geometry can still be used by introducing a four-dimensional reference frame determined by the set of base vectors a 1 , a 2 , a 3 , a 4 with components determined by the rows of the matrix. There many cases that math is the most important part of the life. Mathematics is defined as the science which deals with logic of shape, quantity and arrangement.
During ancient times in Egypt, the Egyptians used math's and complex mathematic equations like geometry and algebra. That is how they managed to build the pyramids. Coordinate systems. When he was a child, he was often sick, so the teachers at his boarding school let him stay in bed until noon. He went on staying in bed until noon for almost all his life.
While in bed, Descartes thought about math and philosophy. One day, Descartes noticed a fly crawling around on the ceiling. He watched the fly for a long time. He wanted to know how to tell someone else where the fly was. Finally he realized that he could describe the position of the fly by its distance from the walls of the room. When he got out of bed, Descartes wrote down what he had discovered. Then he tried describing the positions of points, the same way he described the position of the fly.
Products of vectors.
Descartes had invented the coordinate plane! In fact, the coordinate plane is sometimes called the Cartesian plane, in his honor. History 5. Describing position The position of any object in the real world can be described using a simple coordinate system. The place where the zero values along each axis meet is called the origin. In this example, the X equals 2, Y equals 4, and Z equals 3.
Location of Air Transport. Anytime one has a need to know the location of something — where something should be or where something actually is — a coordinate plane is a very useful tool. For this reason, applications that make use of mapping are common. An air traffic controller must know the location of every aircraft in the sky within certain geographic boundaries.
In order to describe where each aircraft is situated, coordinates are assigned to each vehicle in the air. So coordinate system is one of the most important part of air transport. Map Projections. A projected coordinate system is any coordinate system designed for a flat surface, such as a printed map or a computer screen.
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Both 2D and 3D Cartesian coordinate systems provide the mechanism for describing the geographic location and shape of features using x- and y-values. The Cartesian coordinate system uses two axes: one horizontal x , representing east—west, and one vertical y , representing north—south. The point at which the axes intersect is called the origin. Locations of geographic objects are defined relative to the origin, using the notation x,y , where x refers to the distance along the horizontal axis and y refers to the distance along the vertical axis.
The origin is defined as 0,0. Latitude and longitude Describing the correct location and shape of features requires a coordinate framework for defining real-world locations. A geographic coordinate system is used to assign geographic locations to objects. A global coordinate system of latitude-longitude is one such framework. They are measures of the angles in degrees from the center of the earth to a point on the earth's surface. This type of coordinate reference system is often referred to as a geographic coordinate system. Economy In economics, we use math widely, for analysing and managing.
In economics, the Lorenz curve is a graphical representation of the cumulative distribution function of the empirical probability distribution of wealth or income, and was developed by Max O. Lorenz in for representing inequality of the wealth distribution, and each calculation apply in coordinate system. Military service Cartesian Coordinate system is also important thing in Military Service.
For each target there are coordinates to determine the precise position of them. For example, a soldier want to explode some targets of enemy, so he must know the exact position of the target.
If he know the position with coordinates, then it is so easy to explode or wipe out the target of enemy by warship or others. But it is important that they have to know exact coordinates, because if it is not calibrated correctly it will give rise to untold result. Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their book, Vector Analysis.
In the conventional form using cross products, vector calculus does not generalize to higher dimensions, while the alternative approach of geometric algebra, which uses exterior products does generalize. History Cannon A cannon is any piece of artillery that uses gunpowder or other usually explosive-based propellants to launch a projectile.
Cannon vary in caliber, range, mobility, rate of fire, angle of fire, and firepower; different forms of cannon combine and balance these attributes in varying degrees, depending on their intended use on the battlefield.
Of course for this we need vectors. It is used also in aircraft. The first documented installation of a cannon on an aircraft was on the Voisin Canon in , displayed at the Paris Exposition that year. By World War I, all of the major powers were experimenting with aircraft mounted cannons; however their low rate of fire and great size and weight precluded any of them from being anything other than experimental.
Sports Baseball Another example of a vector in real life would be an outfielder in a baseball game moving a certain direction for a specific distance to reach a high fly ball before it touches the ground. The outfielder can't just run directly for where he sees the ball first or he is going to miss it by a long shot. The player must anticipate what direction and how far the ball will be from him when it drops and move to that location to have the best chance of catching the ball.
Also in basketball match, this is the same. For throwing a ball through a netted hoop, again you have to know the direction or trajectory of ball. Golf is also same, but according to golf ball is small, you must consider the vector of wind force, if wind is strong, you must consider the wind force also in football. Wind Vectors Lets say we have plane with constant velocity, and plane move to south, and we have wind force which direction of it is west, so due to plane movement is south and wind movement is west, finally plane move diagonally, or in the south-west.
There is kind of win that experienced by an observer in motion and is the relative velocity of the wind in relation to the observer is called Apparent wind. Suppose you are riding a bicycle on a day when there is no wind.