Vectors in Two and Three Dimensions
What is a Vector?
In Euclidean geometry, a line segment \(\overline{PQ}\) is defined by two distinct points \(P\) and \(Q\) and it includes all the points on the straight line between these points. The points \(P\) and \(Q\) are called the endpoints. The order of the endpoints does not matter, so the segment \(\overline{PQ}\) is identical to the segment \(\overline{QP}\).
An oriented line segment adds the concept of direction. It is a line segment where one endpoint is designated as the tail (the starting point) and the other is the head (the ending point). The oriented line segment \(\overrightarrow{PQ}\) has tail \(P\) and head \(Q,\) representing movement or displacement specifically from \(P\) to \(Q\).
Two oriented line segments are equivalent if they are parallel, have the same length and the same orientation. Geometrically, oriented line segments \(\overrightarrow{PQ}\) and \(\overrightarrow{ST}\) are equivalent if the quadrilateral \(PQTS\) is a parallelogram. Although they are distinct physical segments, equivalent oriented line segments represent the same relative displacement.
A vector is an abstract concept characterized by a specific magnitude and direction. It is represented visually by any oriented line segment with that magnitude and direction; each such segment is called a representative of the vector. In a coordinate system, the most convenient representative is the one whose tail is at the origin (denoted by O). This particular representative is called the position vector of the point which is its head.
