Consider the following cases: A man walks 200 metres across a field in a direction 45?E of S. A ship steams on a course 30?E of N at a speed of 20 km/h. The wind is blowing at 40 km/h from N. Notice that each motion involves a physical quantity which consists of a magnitude and a specific direction. In order to have a physical description (representation) of the quantity, the notion of a directed line segment is introduced. A directed line segment, denoted by or PQ, is a line segment PQ with the direction along the line segment from P to Q. The directed line segment PQ consists of a magnitude (the length of PQ) and a direction as specified. These characteristics also apply to the physical quantities mentioned above. Each of the above quantities can be represented by a directed line segment as follows: 1. The motion of the man is called a displacement which can be represented by the directed line segments as shown below. This quantity is different from scalar quantities like mass, volume or work done which are specified by magnitude only. This physical quantity is called a vector quantity or simply a vector. It is customary to denote vectors by bold-face letters a, b, c etc. In writing, wavy line letters can be used. Let us denote the above displacement as d. Then we write: or or Notice that we may use different sets of starting points and ending points when representing a vector by a directed line segment. For this reason, this vector is called a free vector. 2. The motion of a ship may be represented by the directed line segment as shown below. In this case, we denote the physical quantity by v and we write: or 3. The motion of the wind may be described by the directed line segments PQ, RS ect as shown below. We may write: v (wind) = or Definition Quantities which have only magnitude and obey the usual laws of algebra are called scalar quantities or scalars. For example, mass, work, work done and energy are scalar quantities. Mathematical quantities which have both magnitude and direction are called vector quantities or simply vectors. Quantities like displacement, velocity, acceleration and force are vectors. Geometrical representation of vectors Geometrically, we specify a vector a with given magnitude and direction as follows: Let P, Q be two points in space such that: (a) the magnitude of a is represented by the length of the line segment PQ (b) the direction of a is indicated by the direction from P to Q. The notation ia thus used to denote the geometrical vector which represents a. Mathematically, if represents a, we may write = a and this is shown below: The modulus of a vector is its magnitude. This the modulus of the vector is written as or . The modulus of a is or simply a.