Physics Notes, Chapter Vectors and Scalars

SCALARS & VECTORS

 Physical quantities which can completely be specified by a number (magnitude) having an appropriate unit are known as "SCALAR QUANTITIES".

Scalar quantities do not need direction for their description. Scalar quantities are comparable only when they have the same physical dimensions. Two or more than two scalar quantities measured in the same system of units are equal if they have the same magnitude and sign. Scalar quantities are denoted by letters in ordinary type. Scalar quantities are added, subtracted, multiplied or divided by the simple rules of algebra.

  Work, energy, electric flux, volume, refractive index, time, speed, electric potential, potential difference, viscosity, density, power, mass, distance, temperature, electric charge, electric flux etc.

Physical quantities having both magnitude and direction with appropriate unit are known as "VECTOR QUANTITIES".

We can't specify a vector quantity without mention of deirection. vector quantities are expressed by using bold letters with arrow sign such as: vector quantities can not be added, subtracted, multiplied or divided by the simple rules of algebra. vector quantities added, subtracted, multiplied or divided by the rules of trigonometry and geometry.

  Velocity, electric field intensity, acceleration, force, momentum, torque, displacement, electric current, weight, angular momentum etc.

  On paper vector quantities are represented by a straight line with arrow head pointing the direction of vector or terminal point of vector. A vector quantity is first transformed into a suitable scale and then a line is drawn with the help of the scale choosen in the given direction.

ADDITION OF VECTORS

Acccording to the parallelogram law of vector addition:

"If two vector quantities are represented by two adjacent sides or a parallelogram then the diagonal of parallelogram will be equal to the resultant of these two vectors."

EXPLANATION

Consider two vectors . Let the vectors have the following orientation

parallelogram of these vectors is :

According to parallelogram law:

  Magintude or resultant vector can be determined by using either sine law or cosine law.

RESOLUTION OF VECTOR

The process of splitting a vector into various parts or components is called "RESOLUTION OF VECTOR" These parts of a vector may act in different directions and are called "components of vector".

We can resolve a vector into a number of components .Generally there are three components of vector viz. Component along X-axis called x-component Component along Y-axis called Y-component Component along Z-axis called Z-component Here we will discuss only two components x-component & Y-component which are perpendicular to each other.These components are called rectangular components of vector.

COMPONENTS

  Consider a vector acting at a point making an angle q with positive X-axis. Vector is represented by a line OA.From point A draw a perpendicular AB on X-axis.Suppose OB and BA represents two vectors.Vector OA is parallel to X-axis and vector BA is parallel to Y-axis.Magnitude of these vectors are V_x and V_y respectively.By the method of head to tail we notice that the sum of these vectors is equal to vector .Thus V_x and V_y are the rectangular components of vector . V_x = Horizontal component of . V_y = Vertical component of .

  Consider right angled triangle DOAB

  Consider right angled triangle DOAB

MULTIPLICATION & DIVISION OF VECTOR BY A NUMBER (SCALAR)

   When a vector is multiplied by a positive number (for example 2, 3 ,5, 60 unit etc.) or a scalar only its magnitude is changed but its direction remains the same as that of the original vector. If however a vector is multiplied by a negative number (for example -2, -3 ,-5, -60 unit etc.) or a scalar not only its magnitude is changed but its direction also reversed.

  The product of a vector by a scalar quantity (m) follows the following rules:

(m) = (m) which is called commutative law of multiplication. m(n) = (mn) which is called associative law of multiplication . (m + n) = m+ n which is called distributive law of multiplication .

DIVISION OF A VECTOR BY A SCALAR

  The division of a vector by a scalar number (n) involves the multiplication of the vector by the reciprocal of the number (n) which generates a new vector. Let n represents a number or scalar and m is its reciprocal then the new vector is given by :

where m = 1/n                              

and its magnitude is given by:

The direction of is same as that of if (n) is a positive number. The direction of is opposite as that of if (n) is a negative number.

ddition of vectors by Head to Tail method (Graphical Method)

  Head to Tail method or graphical method is one of the easiest method used to find the resultant vector of two of more than two vectors.

DETAILS OF METHOD

 

Consider two vectors and acting in the directions as shown below:

In order to get their resultant vector by head to tail method we must follow the following steps:

STEP # 1

  Choose a suitable scale for the vectors so that they can be plotted on the paper.

STEP # 2

  Draw representative line of vector Draw representative line of vector such that the tail of coincides with the head of vector .

STEP # 3

  Join 'O' and 'B'. represents resultant vector of given vectors and i.e.

STEP # 4

  Measure the length of line segment and multiply it with the scale choosen initially to get the magnitude of resultant vector.

STEP # 5

  The direction of the resultant vector is directed from the tail of vector to the head of vector .