Basic Concepts Of Vectors: Vectors And Equilibrium: Learning objectives

At the end of this chapter the students will be able to:

  1. Understand and use rectangular coordinate system.
  2. Understand the idea of unit vector. Null vector and position vector.
  3. Represent a vector as two perpendicular components (rectangular components)
  4. Understand the rule of vector addition and extend it to add vectors using rectangular components.
  5. Understand multiplication of vectors and solve problems.
  6. Define the moment of force or torque.
  7. Appreciate the use of the torque due to a force.
  8. Show an understanding that when there is no resultant force and no resultant torque, a system is in equilibrium.
  9. Appreciate the applications of the principle of moments.
  10. Apply the knowledge gained to solve problems on statics.

Physical quantities that have both numerical and directional properties are called vectors. This chapter is concerned with the vector algebra and its applications in problems of equilibrium of forces and equilibrium of torques.

2.1 Basic Concepts Of Vectors

(i) vectors

As we have studied in school physics, there are some physical quantities which require both magnitude and direction for their complete description, such as velocity, acceleration and force. They are called vectors. In books, vectors are usually denoted by bold face characters such as A, d, r and v while in handwriting, we put an arrowhead over the letter e.g. d. if we which to refer only to the magnitude of a vector d we use light face type such as d.

(ii)  Rectangular Coordinate System

Two reference lines drawn at right angles to each other as shown in Fig. 2.1 (a) are known as coordinate axes and their point of intersection is known as origin. This system of coordinate axes is called Cartesian or rectangular coordinate system.

coordinate_axes

Fig. 2.1 (a)

One of the lines is named as x-axis and the other the y-axis. Usually the x-axis is taken as the horizontal axis, with the positive direction to the right, and the y-axis as the vertical axis with the positive direction upward.

The direction of a vector in a plane is denoted by the angle which the representative line of the vector makes with positive x-axis in the anti-clock wise direction, as shown in Fig 2.1 (b). the point P shown in Fig 2.1 (b) has coordinates (a,b). this notation means that if we start at the origin, we can reach P by moving ‘a’ units along the positive x-axis and then ‘b’ units along the positive y-axis.

positive_x-axis_coordinates_(a,b)

Fig. 2.1 (b)

The direction of a vector in space requires another axis which is at right angle to both x and y axes, as shown in Fig 2.2 (a). the third axis is called z-axis.

right_angle_to both_x and_y axes

Fig. 2.2 (a)

The direction of a vector in space is specified by the three angles which the representative line of the vector makes with x, y and z axes respectively as shown in Fig 2.2 (b). the point P of a vector A is thus denoted by three coordinates (a, b, c).

x, y and z axes_respectively

Fig. 2.2 (b)

(iii) Addition Of Vectors

Given two vectors A and B as shown in Fig 2.3 (a), their sum is obtained by drawing their representative lines in such a way that tail of vector B coincides with the head of the vector A. now if we join the tail of A to the head of B, as shown in the fig. 2.3 (b), the line joining the tail of A to the head of B will represent the vector sum (A+B) in magnitude and direction. The vector sum is also called resultant and is indicated by R. thus R = A+B this is know as head to tail rule of vector addition. This rule can be extended to find the sum of any number of vectors. Similarly the sum B + A is illustrated by black lines in fig 2.3 (c). the answer is same resultant R as indicated by the red line. Therefore, we can say that

addition_of_vectors

A + B = B + A …………..      (2.1)

So the vector addition is said to be commutative. It means that when vectors are added, the result is the same for any order of addition.

(iv) Resultant Vector

The resultant of a number of vectors of the same kind-force vectors for example, is that single vector which would have the same effect as all the original vectors taken together.

(v) Vector Subtraction

The subtraction of a vector is equivalent to the addition of the same vector with its direction reversed. Thus, to subtract vector B from vector A, reverse the direction of B and add it to A, as shown in Fig. 2.3 (d).

subtract_vector_B

Fig 2.3 (d)

AB = A + (-B)                  where (-B) is negative vector of B

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