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A vector is not one thing more than a directed line segment. Technically speaking, a vector is specified as an object that has both a size, or magnitude, and a direction. Vectors have a lot of apps in mathematics and are found abundantly in fields like physics and engineering. Examples of vectors are velocity, acceleration, and force. Vectors are specified by their size and direction. For example, velocity is a vector because we may describe this amount by both it is size, that is speed, and it is direction. Thus a car moving at 60 miles per hour in a direction due north exemplifies a velocity vector. Acceleration likewise typifies a vector because this amount 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 strength of 15 Newtons (on world this is a weight of in regards to 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 may be given as a point. For example, in two dimensions the point (1, 0) represents the vector starting at the origin (we say having it is tail at the origin) pointing to the right and terminating (we say having it is head) one unit from the origin. The point (0, 1) represents the vector having it is tail at the origin, pointing straight up and having it is head one unit from the origin. The point (1, 1) represents the vector having it is head once again at the origin and it is tail at the point (1, 1); this vector lies on the line which bisects the original 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 thence give a vector as (x, y, z). For example, the vector (0,0,1) is that having it is tail at the origin of our three dimensional coordinate system and it is tail one unit straight up from the origin. Similarly, we may give other coordinates to generate vectors pointing in any direction in three space, and such vectors would correspond to such things as strength or velocity in the real world. Once we grasp the basic definition of a vector, we may then talk regarding 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 necessary thing to do not forget is that a vector is plainly a mathematical object that models such real world phenomena as strength and speed. Vectors are directed line segments that may be plotted on a Cartesian Plane using two points in two dimensions, or three points in three dimensions. Other more expansive apps of vectors are studied in courses such as calculus, linear algebra, and physics. |
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