Vectors and Scalars
- Krishna Bahadur Bista
Definitions of vectors and scalars
- The scalars: The quantities which have only magnitudes or amount of matter contained in them are called the vectors. For example; mass of a body, work done, temperature, length, breadth,volume, speed etc. are the scalar quantities. These quantities can be well understood only by their magnitudes as how large or small they are. The scalars are easily added, subtracted,multiplied and divided numerically. So we don't have to learn about them in detail.
- The vectors: The physical quantities which require both the magnitudes and directions to be well understood are called the vector quantities. For example; force, velocity, acceleration, momentum, tension etc. are the vector quantities. If I say, "I am pulling a stone with a 5 N force", this sentence will not be fully understood or clear to the listeners. To make this saying more understandable, I must mention the direction of pulling also as " I am pulling a stone with a 5 N force from east to west or in any other specific direction." Attaching the direction of action makes the introduction of 5 N force more clear. That is why the force is a vector quantity. The vectors appear in the state of motion and the scalars are the quantities which are used to say how large or small the vectors are. In other words they are the magnitudes of the vectors.
- Representation of the vectors: The vectors are represented geometrically by the directed line segments. The arrow head of the line segment shows the direction of the vector. The arrow tail and the head are called the initial point and the final point or the terminal point respectively. The length of the line segment is called the magnitude or the modulus of the vector. The vectors represented by the line segments are shown in the following figure:
Geometrical representation of the vectors
4. Types of vectors: There are various types of vectors defined as below: (i) Zero or Null Vector: A vector whose initial and the terminal points are coincident is called a zero or a null vector. It is also defined as the vector whose magnitude is zero. (ii) Unit Vector: A vector whose magnitude is one or unity is called the unit vector. (iii) Parallel Vectors: The vectors which are represented by the parallel line segments are called the parallel vectors. (iv) Like vectors: The parallel vectors having the same direction are called the like vectors. (v) Unlike vectors: The parallel vectors having just opposite directions are called the unlike vectors. (vi) Equal vectors: The like vectors having equal magnitudes are called the equal vectors. (vii) Position Vector: A vector whose initial point is origin is called the position vector of the terminal point. (viii) Co-initial Vectors: The vectors having the same initial point are called co-initial vectors. (ix) Co-terminal or Co-terminus vectors: The vectors having the same terminal point are called the co-terminal vectors. (x) Negative vector: A vector is called the negative vector of any given vector if its direction is just opposite but the same magnitude as that of a given vector. (xi) Localized Vector: A vector passing through a given point and parallel to the given vector is called a localized vector. (xii) Free Vector: A vector whose position is not fixed or which independent of position is called a free vector. (xiii) Reciprocal Vectors: Two vectors having magnitudes reciprocal of each other are called the reciprocal vectors. (xiv) Collinear Vectors: All the vectors parallel to the same line are called the collinear vectors. All parallel vectors are collinear. (xvi) Coplanar vectors: All the vectors lying on the same or parallel planes are called the coplanar vectors.
Vectors in terms of the coordinates
The vectors are also represented by the coordinates of the end points of the line segments representing the vectors. Let P(x,y) be any point in a plane. Then its position vector is written as vector(O,P) = (x,y). This because (x,y) gives both the length and direction of OP. So is the same vector as represented by segment OP. If P be the point in space with coordinates (x,y,z) then the position vector of P is written as vector(O,P)= (x,y,z). The vector PQ where coordinates of P and Q are (x1,y1) and (x2,y2) is represented by (x2-x1,y2-y1). If P and Q be in space with coordinates (x1,y1,z1) and (x2,y2,z2) the vector represented by the segment PQ is written as (z2-x1,y2-y1,z2-z1). All these vectors are shown in the following figures: