SCALARS AND VECTORS.Physical quantities which can be completely specified by

Scalars and Vectors

Scalars

1.A number which represents the magnitude of the quantity.

2.An appropriate unit are called Scalars.

Scalars quantities can be added, subtracted multiplied and divided by usual algebraic laws.ExamplesMass, distance, volume, density, time, speed, temperature, energy, work, potential, entropy, charge etc.

Vectors

Physical quantities which can be completely specified by

1.A number which represents the magnitude of the quantity.

2.An specific direction

are called Vectors.

Special laws are employed for their mutual operation.

ExamplesDisplacement, force, velocity, acceleration, momentum.

Representation of a VectorA straight line parallel to the direction of the given vector used to represent it. Length of the line on a certain scale specifies the magnitude of the vector. An arrow head is put at one end of the line to indicate the direction of the given vector.

The tail end O is regarded as initial point of vector R and the head P is regarded as the terminal point of the vector R.

Unit VectorA vector whose magnitude is unity (1) and directed along the direction of a given vector, is called the unit vector of the given vector.

A unit vector is usually denoted by a letter with a cap over it. For example if r is the given vector, then r will be the unit vector in the direction of r such that

r = r .r

Orr = r / r

unit vector = vector / magnitude of the vector

Equal VectorsTwo vectors having same directions, magnitude and unit are called equal vectors.

Zero or Null VectorA vector having zero magnitude and whose initial and terminal points are same is called a null vector. It is usually denoted by O. The difference of two equal vectors (same vector) is represented by a null vector.

R - R - O

Free VectorA vector which can be displaced parallel to itself and applied at any point, is known as free vector. It can be specified by giving its magnitude and any two of the angles between the vector and the coordinate axes. In 3-D, it is determined by its three projections on x, y, z-axes.

Position VectorA vector drawn from the origin to a distinct point in space is called position vector, since it determines the position of a point P relative to a fixed point O (origin). It is usually denoted by r. If xi, yi, zk be the x, y, z components of the position vector r, then

r = xi + yj + zk

Negative of a VectorThe vector A. is called the negative of the vector A, if it has same magnitude but opposite direction as that of A. The angle between a vector and its negative vector is always of 180º.

Multiplication of a Vector by a NumberWhen a vector is multiplied by a positive number the magnitude of the vector is multiplied by that number. However, direction of the vector remain same. When a vector is multiplied by a negative number, the magnitude of the vector is multiplied by that number. However, direction of a vector becomes opposite. If a vector is multiplied by zero, the result will be a null vector.

The multiplication of a vector A by two number (m, n) is governed by the following rules.

1.m A = A m

2.m (n A) = (mn) A

3.(m + n) A = mA + nA

4.m(A + B) = mA + mB

Division of a Vector by a Number (Non-Zero)If a vector A is divided by a number n, then it means it is multiplied by the reciprocal of that number i.e. 1/n. The new vector which is obtained by this division has a magnitude 1/n times of A. The direction will be same if n is positive and the direction will be opposite if n is negative.

Resolution of a Vector Into Rectangular Components

DefinitionSplitting up a single vector into its rectangular components is called the Resolution of a vector.

Rectangular Components

Components of a vector making an angle of 90º with each other are called rectangular components.

ProcedureLet us consider a vector F represented by OA, making an angle O with the horizontal direction.

Draw perpendicular AB and AC from point on X and Y axes respectively. Vectors OB and OC represented by Fx and Fy are known as the rectangular components of F. From head to tail rule of vector addition.

OA = OB + BA

F = Fx + Fy

To find the magnitude of Fx and Fy, consider the right angled triangle OBA.

Fx / F = Cos ? => Fx = F cos ?

Fy / F = sin ? => Fy = F sin ?

Addition of Vectors by Rectangular ComponentsConsider two vectors A1 and A2 making angles ?1 and ?2 with x-axis respectively as shown in figure. A1 and A2 are added by using head to tail rule to give the resultant vector A.

The addition of two vectors A1 and A2 mentioned in the above figure, consists of following four steps.

Step 1For the x-components of A, we add the x-components of A1 and A2 which are A1x and A2x. If the x-components of A is denoted by Ax then

Ax = A1x + A2x

Taking magnitudes only

Ax = A1x + A2x

OrAx = A1 cos ?1 + A2 cos ?2 ................. (1)

Step 2For the y-components of A, we add the y-components of A1 and A2 which are A1y and A2y. If the y-components of A is denoted by Ay then

Ay = A1y + A2y

Taking magnitudes only

Ay = A1y + A2y

OrAy = A1 sin ?1 + A2 sin ?2 ................. (2)

Step 3Substituting the value of Ax and Ay from equations (1) and (2) respectively in equation (3) below, we get the magnitude of the resultant A

A = |A| = v (Ax)2 + (Ay)2 .................. (3)

Step 4By applying the trigonometric ratio of tangent ? on triangle OAB, we can find the direction of the resultant vector A i.e. angle ? which A makes with the positive x-axis.

tan ? = Ay / Ax

? = tan-1 [Ay / Ax]

Here four cases arise

(a)If Ax and Ay are both positive, then

? = tan-1 |Ay / Ax|

(b)If Ax is negative and Ay is positive, then

? = 180º - tan-1 |Ay / Ax|

( c)If Ax is positive and Ay is negative, then

? = 360º - tan-1 |Ay / Ax|

(d)If Ax and Ay are both negative, then

? = 180º + tan-1 |Ay / Ax|

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