The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages. 
Home Vectors Vector Algebra Vectors  
Search the VIAS Library  Index  
Vectors
Usually we are not really interested in the exact placement of a directed line segment on the (x, y) plane, but in the length and direction of . These can be determined by the x and y components of . We are thus led to the notion of a vector. DEFINITION The family of all directed line segments with the same components as will be called the vector from P to Q. We say that represents this vector. Since all directed line segments with the same components have the same length and direction, a vector may be regarded as a quantity which has length and direction. Vectors arise quite naturally in both physics and economics. Here are some examples of vector quantities.
We shall now begin the algebra of vectors. In vector algebra, real numbers are called scalars. We study two different kinds of quantities, scalars and vectors. The length (or norm) of a vector A is the distance between P and Q where represents A. The length is a scalar, denoted by A. If A has components a_{1} and a_{2}, then the length, shown in Figure 10.1.5, is given by the distance formula, The length of a position vector is the distance from the origin. The length of a Figure 10.1.5 Length of a Vector velocity vector is the speed of a particle. The length of a force vector is the magnitude of the force. The length of a displacement vector is the distance moved. For price or commodity vectors, the notion of length does not arise in a natural way.
The vector with components (0, 0) is called the zero vector, denoted by 0. The zero vector is represented by the degenerate line segments . It has no direction. The length of the zero vector is zero, while the length of every other vector is a positive scalar.


Home Vectors Vector Algebra Vectors 