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Article Source: http://EzineArticles.com/6100228segment. Technically speaking, a vector is defined as an object that has both a size, or magnitude, and a direction. Vectors have many applications in mathematics and are found abundantly in fields like physics and engineering. Examples of vectors are velocity, acceleration, and force.
Vectors are defined by their size and direction. For example, velocity is a vector because we can describe this quantity by both its size, that is speed, and its direction. Thus a car moving at 60 miles per hour in a direction due north exemplifies a velocity vector. Acceleration also typifies a vector because this quantity has both a size and direction. A truck accelerating at 10 feet per second per second moving due south exemplifies an acceleration vector. Force is another object which is modeled by a vector quantity. A force of 15 Newtons (on earth this is a weight of about 3lbs) exerted downward is an example of a vector.
In the Cartesian Plane---the grid on which we graph points, lines, and curves---a vector can be given as a point. For example, in two dimensions the point (1, 0) represents the vector starting at the origin (we say having its tail at the origin) pointing to the right and terminating (we say having its head) one unit from the origin. The point (0, 1) represents the vector having its tail at the origin, pointing straight up and having its head one unit from the origin. The point (1, 1) represents the vector having its head once again at the origin and its tail at the point (1, 1); this vector lies on the line which bisects the first quadrant.
Since we live in a three dimensional world, we need to introduce vectors in three-space. These are analogous to those given in two-space, except that now we use three values. That is, we specify the x, y, and z coordinates and thus give a vector as (x, y, z). For example, the vector (0,0,1) is that having its tail at the origin of our three dimensional coordinate system and its tail one unit straight up from the origin. Similarly, we can give other coordinates to generate vectors pointing in any direction in three space, and such vectors would correspond to such things as force or velocity in the real world.
Once we understand the basic definition of a vector, we can then talk about operations with them: these operations include addition, subtraction, and a special kind of multiplication called scalar multiplication. Such operations would come into play when, for example, combining (adding) or subtracting forces or accelerations.
The most important thing to remember is that a vector is simply a mathematical object that models such real world phenomena as force and speed. Vectors are directed line segments that can be plotted on a Cartesian Plane using two points in two dimensions, or three points in three dimensions. Other more expansive applications of vectors are studied in courses such as calculus, linear algebra, and physics.
A vector is nothing more than a directed line segment. Technically speaking, a vector is defined as an object that has both a size, or magnitude, and a direction. Vectors have many applications in mathematics and are found abundantly in fields like physics and engineering. Examples of vectors are velocity, acceleration, and force.
Vectors are defined by their size and direction. For example, velocity is a vector because we can describe this quantity by both its size, that is speed, and its direction. Thus a car moving at 60 miles per hour in a direction due north exemplifies a velocity vector. Acceleration also typifies a vector because this quantity has both a size and direction. A truck accelerating at 10 feet per second per second moving due south exemplifies an acceleration vector. Force is another object which is modeled by a vector quantity. A force of 15 Newtons (on earth this is a weight of about 3lbs) exerted downward is an example of a vector.
In the Cartesian Plane---the grid on which we graph points, lines, and curves---a vector can be given as a point. For example, in two dimensions the point (1, 0) represents the vector starting at the origin (we say having its tail at the origin) pointing to the right and terminating (we say having its head) one unit from the origin. The point (0, 1) represents the vector having its tail at the origin, pointing straight up and having its head one unit from the origin. The point (1, 1) represents the vector having its head once again at the origin and its tail at the point (1, 1); this vector lies on the line which bisects the first quadrant.
Since we live in a three dimensional world, we need to introduce vectors in three-space. These are analogous to those given in two-space, except that now we use three values. That is, we specify the x, y, and z coordinates and thus give a vector as (x, y, z). For example, the vector (0,0,1) is that having its tail at the origin of our three dimensional coordinate system and its tail one unit straight up from the origin. Similarly, we can give other coordinates to generate vectors pointing in any direction in three space, and such vectors would correspond to such things as force or velocity in the real world.
Once we understand the basic definition of a vector, we can then talk about operations with them: these operations include addition, subtraction, and a special kind of multiplication called scalar multiplication. Such operations would come into play when, for example, combining (adding) or subtracting forces or accelerations.
The most important thing to remember is that a vector is simply a mathematical object that models such real world phenomena as force and speed. Vectors are directed line segments that can be plotted on a Cartesian Plane using two points in two dimensions, or three points in three dimensions. Other more expansive applications of vectors are studied in courses such as calculus, linear algebra, and physics.
Vectors are defined by their size and direction. For example, velocity is a vector because we can describe this quantity by both its size, that is speed, and its direction. Thus a car moving at 60 miles per hour in a direction due north exemplifies a velocity vector. Acceleration also typifies a vector because this quantity has both a size and direction. A truck accelerating at 10 feet per second per second moving due south exemplifies an acceleration vector. Force is another object which is modeled by a vector quantity. A force of 15 Newtons (on earth this is a weight of about 3lbs) exerted downward is an example of a vector.
In the Cartesian Plane---the grid on which we graph points, lines, and curves---a vector can be given as a point. For example, in two dimensions the point (1, 0) represents the vector starting at the origin (we say having its tail at the origin) pointing to the right and terminating (we say having its head) one unit from the origin. The point (0, 1) represents the vector having its tail at the origin, pointing straight up and having its head one unit from the origin. The point (1, 1) represents the vector having its head once again at the origin and its tail at the point (1, 1); this vector lies on the line which bisects the first quadrant.
Since we live in a three dimensional world, we need to introduce vectors in three-space. These are analogous to those given in two-space, except that now we use three values. That is, we specify the x, y, and z coordinates and thus give a vector as (x, y, z). For example, the vector (0,0,1) is that having its tail at the origin of our three dimensional coordinate system and its tail one unit straight up from the origin. Similarly, we can give other coordinates to generate vectors pointing in any direction in three space, and such vectors would correspond to such things as force or velocity in the real world.
Once we understand the basic definition of a vector, we can then talk about operations with them: these operations include addition, subtraction, and a special kind of multiplication called scalar multiplication. Such operations would come into play when, for example, combining (adding) or subtracting forces or accelerations.
The most important thing to remember is that a vector is simply a mathematical object that models such real world phenomena as force and speed. Vectors are directed line segments that can be plotted on a Cartesian Plane using two points in two dimensions, or three points in three dimensions. Other more expansive applications of vectors are studied in courses such as calculus, linear algebra, and physics.
Joe is a prolific writer of self-help and educational material and is the creator and author of over a dozen books and ebooks which have been read throughout the world. He is a former teacher of high school and college mathematics and has recently returned as a professor of mathematics at a local community college in New Jersey. Joe propagates his Wiz Kid Teaching Philosophy through his writings and lectures and loves to turn "math-haters" into "math-lovers." See his website http://www.mathbyjoe.com/ for more information and for testimonials, and try out one of his ebooks here http://www.mathbyjoe.com/page/page/2924777.htm to achieve better grades in math. Article Source: http://ezinearticles.com/?expert=Joe_Pagano |
Article Source: http://EzineArticles.com/6100228
Vectors are defined by their size and direction. For example, velocity is a vector because we can describe this quantity by both its size, that is speed, and its direction. Thus a car moving at 60 miles per hour in a direction due north exemplifies a velocity vector. Acceleration also typifies a vector because this quantity has both a size and direction. A truck accelerating at 10 feet per second per second moving due south exemplifies an acceleration vector. Force is another object which is modeled by a vector quantity. A force of 15 Newtons (on earth this is a weight of about 3lbs) exerted downward is an example of a vector.
In the Cartesian Plane---the grid on which we graph points, lines, and curves---a vector can be given as a point. For example, in two dimensions the point (1, 0) represents the vector starting at the origin (we say having its tail at the origin) pointing to the right and terminating (we say having its head) one unit from the origin. The point (0, 1) represents the vector having its tail at the origin, pointing straight up and having its head one unit from the origin. The point (1, 1) represents the vector having its head once again at the origin and its tail at the point (1, 1); this vector lies on the line which bisects the first quadrant.
Since we live in a three dimensional world, we need to introduce vectors in three-space. These are analogous to those given in two-space, except that now we use three values. That is, we specify the x, y, and z coordinates and thus give a vector as (x, y, z). For example, the vector (0,0,1) is that having its tail at the origin of our three dimensional coordinate system and its tail one unit straight up from the origin. Similarly, we can give other coordinates to generate vectors pointing in any direction in three space, and such vectors would correspond to such things as force or velocity in the real world.
Once we understand the basic definition of a vector, we can then talk about operations with them: these operations include addition, subtraction, and a special kind of multiplication called scalar multiplication. Such operations would come into play when, for example, combining (adding) or subtracting forces or accelerations.
The most important thing to remember is that a vector is simply a mathematical object that models such real world phenomena as force and speed. Vectors are directed line segments that can be plotted on a Cartesian Plane using two points in two dimensions, or three points in three dimensions. Other more expansive applications of vectors are studied in courses such as calculus, linear algebra, and physics.
Joe is a prolific writer of self-help and educational material and is the creator and author of over a dozen books and ebooks which have been read throughout the world. He is a former teacher of high school and college mathematics and has recently returned as a professor of mathematics at a local community college in New Jersey. Joe propagates his Wiz Kid Teaching Philosophy through his writings and lectures and loves to turn "math-haters" into "math-lovers." See his website http://www.mathbyjoe.com/ for more information and for testimonials, and try out one of his ebooks here http://www.mathbyjoe.com/page/page/2924777.htm to achieve better grades in math. Article Source: http://ezinearticles.com/?expert=Joe_Pagano |
Article Source: http://EzineArticles.com/6100228
