Introduction
Summary In \(\RNrSpc{n}\) we introduce the concepts of arrow and vector as a means to express relationships between points.
Outline Let us begin with arrows: Suppose you move from a point \(P\) in \(\RNrSpc{n}\) to another point, say \(Q\). Then this passage from \(P\) to \(Q\) may be described by the
arrow whose tail is at
\(P\) and whose tip is at \(Q\). For example in \(\RNrSpc{2}\), the passage from \(P(-4,1)\) to \(Q(2,3)\) is
expressed by the arrow \(\Arrow{P}{Q}\).
Vectors When we move an object in space, we need one arrow for each point of the object to describe its location before and after the move. Now, comes a key observation: if the movement was just a
shift, then all of these arrows have the same length and the same direction. As these arrows serve a common purpose, we bundle them all together into a new object, called a
vector. Next, let us turn to the question: how do we determine when two arrows belong to the same vector?
Interplay between vectors and arrows To answer this question, consider two arrows, say \(\Arrow{A}{B}\) and \(\Arrow{X}{Y}\), joining \(A(a_1,\dots ,a_n)\) to \(B(b_1,\dots ,b_n)\) and joining \(X(x_1,\dots ,x_n)\) to \(Y(y_1,\dots ,y_n)\). The instructions for passing from \(A\) to \(B\) are given by the \(n\)-tuple \((b_1-a_1,\dots ,b_n-a_n)\), while the instructions for passing from \(X\) to \(Y\) are given by the \(n\)-tuple \((y_1-x_1,\dots ,y_n-x_n)\). The essential observation here is that \(\Arrow{A}{B}\) and \(\Arrow{X}{Y}\) have the same length and direction exactly when their tail-to-tip instructions are equal; i.e. when
\[(b_1-a_1,\dots ,b_n-a_n) = (y_1-x_1,\dots ,y_n-x_n)\]
Accordingly, we call two
arrows equivalent
if their tail-to-tip instructions match, and we take the collection of all arrows with a given \(n\)-tuple \((v_1,\dots ,v_n)\) of
tail-to-tip instructions
to be a
vector, let's call it \(\Vect{v}\). Since the \(n\)-tuple \((v_1,\dots ,v_n)\) and \(\Vect{v}\) determine each other, we may equate them and write \(\Vect{v}=(v_1,\dots ,v_n)\).
An \(n\)-tuple of numbers is thus doing
double duty. For example, the 2-tuple \((3,2)\) may be used to give the location of a point in \(\RNrSpc{2}\). But it may also be used for the vector whose arrows are such that the passage from their tail to their trip goes 3 units in the \(x\)-direction and, from there, 2 units in the \(y\)-direction. Unfortunately, adding to such confusion is the standard practice to introduce a vector \(\Vect{x}=(x_1,\dots ,x_n)\) and then call it a point; similarly to introduce a point \(P(p_1,\dots ,p_n)\) and then refer to it as a vector. The only justification for this ambiguity is the resulting economy of speech:
- Given point \(P(p_1,\dots ,p_n)\), if we call \((p_1,\dots ,p_n)\) a vector, we mean the vector \(\Vect{p}\) represented by the arrow whose tail is at the origin and whose tip is at \(P\).
-
Given vector \(\Vect{x}=(x_1,\dots ,x_n)\), if we refer to it as a point, we mean \(X(x_1,\dots ,x_n)\), that is the point at the tip of the arrow whose tail is at the origin and which represents \(\Vect{x}\). In this case we say that \(\Vect{x}\) is the position vector for the point \(X\).
Often one is interested in the length of an arrow. Here, our formula for the distance between two points allows us to
answer such a problem immediately. For example, any arrow representing any of the
basic coordinate vectors
of \(\RNrSpc{n}\) has length 1.
Generalizations The basic view of vectors developed here lends itself to more general interpretations. First, in the natural sciences, it common practice to model any quantity which is determined by its direction and its magnitude by a vector. Velocity and acceleration are examples. More abstractly, certain sets may be equipped with operations which make their elements behave like vectors. Such spaces are called ‘vector spaces’ and are studied in the theory of linear algebra, of functional analysis and elsewhere.
Now to the details.