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Unformatted text preview: 2 Traditional Vector Theory The earliest definition of a “vector” usually encountered is that a vector is a thing possessing “length” and “direction”. This is the “arrow in space” view with “length” naturally being the length of the arrow and “direction” being the direction the arrow is pointing. We will denote vectors by writing their name in boldface (e.g., v and a ) or by drawing a little arrow above their name (e.g., −→ v and −→ a ). Some examples of such vectors include the velocity of an object at a particular time and the acceleration of an object at a particular time. (For now, we are not considering vector fields; that is, our vectors will not be functions of time or position.) Perhaps the most fundamental of vectors are those describing “relative position” or “dis- placement”: If A and B are two points in space (i.e., positions in space), then the vector from A to B , denoted −→ AB , is simply the arrow 1 starting at point A and ending at point B . −→ AB gives you the direction and distance to move from position A to position B (hence the term “relative position”). We will use “displacement vectors” as the basic model for traditional vectors. Almost all other traditional vectors in physics — velocities, accelerations, forces, etc. — are derived from displacement vectors. For example, the velocity v of an object at a given position p is the limit v = lim Delta1 t → −→ pq Delta1 t where q is the position of the object Delta1 t time units after being at p . It is through these derivations that the properties we derive for displacement (relative position) vectors can be shown to hold for all the traditional vectors in physics. By the way, throughout this discussion, we are assuming that the points in space are points in a “Euclidean” space. We will discuss exactly what this means later (in another chapter). For now, “assume high school geometry”. In particular: 1. Any two points can be connected by a straight line segment (totally contained in the space) whose length is the distance between the two points. 2. The angles of each triangle add up to π (or 180 degrees). 3. The laws of similar triangles hold. 4. Parallelograms are well defined. 1 officially, −→ AB is a “directed line segment” . 8/25/2011 Traditional Vector Theory Chapter & Page: 2–2 Keep in mind that there are different Euclidean spaces. Two different “flat” planes, for example, are two different Euclidean spaces. ? ◮ Exercise 2.1: Give several reasons why a sphere is not a Euclidean space. 2 2.1 Fundamental Defining Concepts Fundamental Geometrically Defined Concepts The Two Fundamental Measurable Quantities Our goal is to develop the fundamental theory of vectors as things that are completely defined by “length” and “direction”, using the set of vectors describing relative position in some Euclidean space as a basic model. Keep in mind the requirement that “length” and “direction” completelyspace as a basic model....
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