5.8. PHYSICAL VECTORS 67

5.8 Physical VectorsSuppose you push on something. What is important? There are really two things which areimportant, how hard you push and the direction you push.

Definition 5.8.1 Force is a vector. The magnitude of this vector is a measure of how hardit is pushing. It is measured in units such as Newtons or pounds or tons. Its direction is thedirection in which the push is taking place.

Of course this is a little vague and will be left a little vague until the presentation ofNewton’s second law later. See the appendix on this or any physics book.

Vectors are used to model force and other physical vectors like velocity. What was justdescribed would be called a force vector. It has two essential ingredients, its magnitudeand its direction. Think of vectors as directed line segments or arrows as shown in thefollowing picture in which all the directed line segments are considered to be the samevector because they have the same direction, the direction in which the arrows point, andthe same magnitude (length).

Because of this fact that only direction and magnitude are important, it is always possi-ble to put a vector in a certain particularly simple form. Let −→pq be a directed line segmentor vector. Then from Definition 5.4.4 it follows that −→pq consists of the points of the form

p+ t (q−p)

where t ∈ [0,1] . Subtract p from all these points to obtain the directed line segment con-sisting of the points

0+ t (q−p) , t ∈ [0,1] .

The point in Rn,q−p, will represent the vector.Geometrically, the arrow −→pq, was slid so it points in the same direction and the base is

at the origin 0. For example, see the following picture.

In this way vectors can be identified with points of Rn.

Definition 5.8.2 Let x= (x1, · · · ,xn) ∈ Rn. The position vector of this point is the vectorwhose point is at x and whose tail is at the origin (0, · · · ,0). If x= (x1, · · · ,xn) is called