The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.


Usually we are not really interested in the exact placement of a directed line segment 10_vectors-11.gif on the (x, y) plane, but in the length and direction of 10_vectors-12.gif. These can be determined by the x and y components of 10_vectors-13.gif. We are thus led to the notion of a vector.


The family of all directed line segments with the same components as 10_vectors-14.gif will be called the vector from P to Q. We say that 10_vectors-15.gif 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.


If an object is at the point (p1,p2) in the plane, its position vector is the vector with components p1 and p2.


If a particle is moving in the plane according to the parametric equations x = f(t), y = g(t), the velocity vector is the vector with x and y components dx/dt and dy/dt.


The acceleration vector of a moving particle has the x and y components d2x/dt2 and d2y/dt2.


In physics, force is a vector quantity which will accelerate a free particle in the direction of the force vector at a rate proportional to the length of the force vector

Displacement (change in position)

If an object moves from the point P to the point Q, its displacement vector is the vector from P to Q.

Commodity vector

In economics, one often compares two or more commodities (such as guns and butter). If a trader in a market has a quantity a1 of one commodity and a2 of another, his commodity vector has the x and y components (a1, a2).

Price vector

If two commodities have prices p1 and p2 respectively, the price vector has components (p1, p2). The components of a commodity or price vector are always greater than or equal to zero.

Example 1
Example 2

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 10_vectors-21.gif represents A. The length is a scalar, denoted by |A|. If A has components a1 and a2, 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.

Example 3

The vector with components (0, 0) is called the zero vector, denoted by 0. The zero vector is represented by the degenerate line segments 10_vectors-26.gif. It has no direction. The length of the zero vector is zero, while the length of every other vector is a positive scalar.

Last Update: 2006-11-07