Vectors: The notion of vector and vector space
Vectors in a plane or space
Scalar or vector? Physical quantities such as mass, height, and temperature have values that are specified by a single value in suitable units. Such quantities, which that have only magnitude, are called scalar quantities. With other variables you need more numbers: for example, think of position where you need two or three numbers, namely the coordinates relative to a certain coordinate system. A position in a geometric plane or space is also determined by the distance to the origin and the direction from the origin to the given position. Physical quantities are called vector quantities when they have besides a magnitute also a direction. Other examples of vector quantities are speed, force, and electric field.
Notation for vectors Typically, vector quantities are written in books in bold or are denoted by symbols with a horizontally and to the right directed arrow above it, such as in \(\mathbf{a}\) for acceleration and \(\vec{F}\) for force.
We will use the arrow notation because this format can be handled with pencil and paper.
We illustrate the notion of vector with a special case, namely vectors in a geometric plane. With a geometric plane we mean a straight plane and we denote it by \(\mathbb{E}^2\). We distinguish this plane form the coordinate plane \(\mathbb{R}^2 = \{(x,y) \mid x\in \mathbb{R}\text{ en }y\in \mathbb{R}\}\), in which a coordinate system has been chosen. A point in the geometric plane is not equal to a set of coordinates. But we need the concept of length in the geometric plane: mathematicians use the term norm instead of length and talk about a normed space.
Vectors in a plane A free vector, or vector in short, is defined as an arrow in the plane with a certain direction and length. In other words, a vector is an oriented segment between two points in the plane with a direction; the position of the vector in the plane is not relevant, that is, it can be placed anywhere without changing its orientation. The equivalence class of arrows is all that matters.
The norm or length of the vector is the length of the segment. The length of a vector \(\vec{v}\) is usually denoted by \(|\vec{v}|\) and sometimes simply by \(v\).
When we drag a vector so that its initial point will lie elsewhere (but the direction and length remain unchanged), we may consider the new arrow as a representative of the same vector.
\(\vec{PQ}\) denotes the representative of a vector from the initial point \(P\) to the terminal point or head \(Q\).
Below we consider a geometric plane with a fixed origin \(O\). In the figure we have drawn a green position vector \(\vec{v}\), which you can modify by dragging the dot at the end of the arrow. In addition, some vectors have been drawn in blue thath are representatives of \(\vec{v}\).
The zero vector There is one special vector, namely a vector of length \(0\). This vector has no direction. \(\vec{0}\) denotes the zero vector.
If \(P\) is a point in the plane, then the vector \(\vec{PP}\) represents the zero vector.
Correspondence between vector and point Once an origin has been selected in a geometric plane it is common practice to let vectors start in the origin. The vector is then uniquely determined by its terminal point, so that vectors can be considered as points in the plane: the point \(P\) is then identified with the vector \(\vec{OP}\). If we want to be very precise we speak in this case about a position vector. Sometimes we abbreviate the vector notation \(\vec{OP}\) for a point \(P\) to \(\vec{P}\). The line through the origin and the point \(P\) is also called the span of the vector \(\vec{OP}\). We also denote this span as \(\langle\vec{P}\rangle\).
The length of the position vector \(\vec{OP}\) is equal to the distance between the points \(O\) and \(P\).
