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Vectors Basics

[vc_row][vc_column width=”2/3″][vc_column_text]Vector analysis is a mathematical subject which is much better taught by mathematicians than by engineers. Vector analysis is mathematical shorthand. It has some new symbols, some new rules, and a pitfall here and there like most new fields, and it demands concentration, attention, and practice.

SCALARS AND VECTORS

Scalar

  • The term scalar refers to a quantity whose value may be represented by a single (positive or negative) real number.
  • The x, y, and z we use in basic algebra are scalars, and the quantities they represent are scalars.

Examples: –

  • If we speak of a body falling a distance L in a time t or the temperature T at any point in a bowl of soup whose coordinates are x, y, and z, then L, t, T, x, y, and z are all scalars.
  • Mass
  • Density
  • pressure (but not force)
  • volume
  • volume resistivity
  • Voltage is also a scalar quantity, although the complex representation of a sinusoidal voltage, an artificial procedure, produces a complex scalar, or phasor, which requires two real numbers for its representation, such as amplitude and phase angle, or real part and imaginary part.

Vector

  • A vector quantity has both a magnitude and a direction in space.

Examples: –

  • Force
  • Velocity
  • Acceleration
  • A straight line from the positive to the negative terminal of a storage battery.

Field

  • A field (scalar or vector) may be defined mathematically as some function of that vector which connects an arbitrary origin to a general point in space.

We usually find it possible to associate some physical effect with a field, such as the force on a compass needle in the earth’s magnetic field, or the movement of smoke particles in the field defined by the vector velocity of air in some region of space. Note that the field concept invariably is related to a region.

Some quantity is defined at every point in a region. Both scalar fields and vector fields exist.

Examples of Scalar Field

  • The temperature throughout the bowl of soup
  • The density at any point in the earth

Examples of Vector Field

  • The gravitational and magnetic fields of the earth
  • The voltage gradient in a cable
  • The temperature gradient in a soldering- iron tip.

The value of a field varies in general with both position and time.

VECTOR ALGEBRA

In Boolean algebra the product AB can be only unity or zero. Vector algebra has its own set of rules, and we must be constantly on guard against the mental forces exerted by the more familiar rules or scalar algebra.

Vectorial addition

  • Vectorial addition follows the parallelogram law, and this is easily, if inaccurately, accomplished graphically.
  • Fig. 1 shows the sum of two vectors, A and B. It is easily seen that

or that vector addition obeys the com- mutative law. Vector addition also obeys the associative law,

Note that when a vector is drawn as an arrow of finite length, its location is defined to be at the tail end of the arrow.

Coplanar vectors, or vectors lying in a common plane, such as those shown in Fig. 1, which both lie in the plane of the paper, may also be added by expressing each vector  in terms of horizontal and vertical components and adding the corresponding components.

Figure 1

Vectors in three dimensions may likewise be added by expressing the vectors in terms of three components and adding the corresponding components.

Vectorial subtraction

  • The rule for the subtraction of vectors follows easily from that for addition, for we may always express

the sign, or direction, of the second vector is reversed, and this vector is then added to the first by the rule for vector addition.

Vectorial multiplication

  • Vectors may be multiplied by scalars.

  • The magnitude of the vector changes, but its direction does not when the scalar is positive, although it reverses direction when multiplied by a negative scalar.
  • Multiplication of a vector by a scalar also obeys the associative and distributive laws of algebra, leading to

Vectorial division

  • Division of a vector by a scalar is merely multiplication by the reciprocal of that scalar.

Two vectors are said to be equal if their difference is zero, or A = B if A – B = O.[/vc_column_text][/vc_column][vc_column width=”1/3″][/vc_column][/vc_row][vc_row][vc_column width=”2/3″][vc_column_text]AUTHORS
1.Bunty B. Bommera
2.Dakshata U. Kamble[/vc_column_text][/vc_column][vc_column width=”1/3″][/vc_column][/vc_row]

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