Vector Resolution

A vector will be generally having components in different directions like x,y and z. Dividing the vector into components along this directions is called resolution of vectors.

We are able to identify value of vectors along the corresponding directions with the help of components of vectors.

Let a and b be any two non-zero vectors in a plane with different directions and let A be another vector in the same plane. A can be expressed as a sum of two vectors – one obtained by multiplying a by a real number and the other obtained by multiplying b by another real number.

To see this, let O and P be the tail and head of the vector A. Then, through O, draw a straight line parallel to a, and through P, a straight line parallel to b. Let them intersect at Q. Then, we have

A = OP = OQ + QP

But since OQ is parallel to a, and QP is parallel to b, we can write :

OQ = λ a, and QP = µ b where λ and µ are real numbers.

Therefore, A = λ a + µ b.

We say that A has been resolved into two component vectors λ a and μ b along a and b respectively. Using this method one can resolve a given vector into two component vectors along
a set of two vectors – all the three lie in the same plane. It is convenient to resolve a general vector along the axes of a rectangular coordinate system using vectors of unit magnitude.

Unit vectors: A unit vector is a vector of unit magnitude and points in a particular direction. It has no dimension and unit. It is used to specify a direction only. Unit vectors along the x-, y and z-axes of a rectangular coordinate system are denoted by iˆ , jˆ and kˆ , respectively, as shown in Figure below.

These unit vectors are perpendicular to each other.

If we multiply a unit vector, say n by a scalar, the result is a vector λ = λ n . In general, a vector A can be written as A = |A|n.

Vector resolution in two dimensions basing on Unit vectors :

Consider a vector A that lies in x-y plane as shown in Figure below. We draw lines from the head of A perpendicular to the coordinate axes and get vectors A1 and A2 such that A1 + A2 = A. Since A1 is parallel to I and A2 is parallel to J , we have :

A1 = Ax i, A2 = Ay j where Ax and Ay are real numbers.

So we can represent the vector as shown.

A = Ax i+ Ay j

Using simple trigonometry, we can express Ax and Ay in terms of the magnitude of A and the angle θ it makes with the x-axis :

Ax = A cos θ
Ay = A sin θ

As is clear from Eq. a component of a vector can be positive, negative or zero depending on the value of θ.

Now, we have two ways to specify a vector A in a plane. It can be specified by :

(i) its magnitude A and the direction θ it makes with the x-axis; or

(ii) its components Ax and Ay If A and θ are given, Ax and Ay can be obtained using Eq. If Ax and Ay are given, A and θ. Then we can deduce the following relations .

The previous topics of vectors can be browsed here below.

Vectors Cross Product
Vectors Dot Product
Concepts of vectors part one and two

Discovery of Neutron,Radius and Density of Nucleus 

Mass Defect and Binding Energy of Nucleus

Radio Activity and Half Life Period

Nuclei Complete Lesson

Vectors Cross Product

The previous post of the blog deals with dot product of vectors.Cross product is another way of multiplying two vectors . Here the the result of product is a vector which will have both magnitude and direction.

1 . When the perpendicular component of one vector with respect to the another vector is effective then the cross product is taken.

2 . The cross product of two vectors is a vector and its direction is given by right hand cork screw rule.

3 . If a and b are two vectors and the angle between them is then the cross product of and is given by a×b = |a| |b| sin Ø( n) where n s a unit vector perpendicular to the plane containing a and b .

4 . If two vectors are parallel i.e. Θ = 0 or 180 then a × b = 0 .

5 . If two vectors are perpendicular to each other a × b = ab and it is maximum .

6 . If i , j and k are unit vectors then

APPLICATIONS OF CROSS PRODUCT OF VECTORS :

RELATED POST

BASICS OF VECTORS PART ONE AND TWO.
SCALAR PRODUCT OF VECTORS

Discovery of Neutron,Radius and Density of Nucleus 

Mass Defect and Binding Energy of Nucleus

Radio Activity and Half Life Period

Nuclei Complete Lesson

Dot Product of Vectors

The previous post of the vector topic is regarding parallelogram law and definition of different kinds of vectors.Here we are going to discuss product of vectors.Here there are three possibilities.

1. Vector multiplied with scalar gives a resultant of vector.
2. Vector multiplied with vector gives a resultant of scalar(Dot Product)
3. Vector multiplied with vector gives a resultant of vector(Cross Product)

Here is the explanation in detail for each time of multiplication.

CASE ONE :

1. When a vector is multiplied by a scalar its products is a vector whose magnitude is equal to the scalar times the magnitude of the given vector.

2.The direction of a vector is same as the given vector, if the scalar is positive and opposite if the scalar is negative.

Example :

P = m v where P is momentum, m is mass and v is velocity.
F = m a where F is force , m is mass and a is acceleration.

NOTE :

A vector multipllied by another vector may give a scalar (or) a vector. Hence there are two types of products for multiplication of two types of products for multiplication of two vectors.

a) dot product (or) Scalar product

b) cross product (or) vector product

CASE TWO SCALAR PRODUCT (OR) DOT PRODUCT PROPERTIES :

1 . When the magnitude of one vector along another vector is effective then the dot product of two vectors is taken.

2. The dot product of two vectors is a scalar.

3. The scalar product of two vectors and is a.b = ab cos θ

4. Scalar product is commutative i.e. a.b = b.a

5 . Scalar product is distributive i.e a.(b+c) = a.b + a.c

6 . The scalar product of two parallel vectors is maximum I.e when θ = 0

7 . The scalar product of two opposite vectors is negative i.e when θ = 180.

8 . The scalar product of two perpendicular vectors is zero when θ = 90.

9 . In case of unit vectors i.i = j.j = k.k = 1 i.e i.i = 1*1*cos 0 = 1

10 . Similarly i.j = j.k = k.i = 0 since i.j = 1 *1* cos 90 = 0.

11 . In terms of Components A.B = A_xB_x + A_yxB_y + A_zB_z .

APPLICATIONS OF DOT PRODUCT :

1 . W = F.S Dot product of force and displacement is work .

2 . P = F.V Dot product of force and velocity is power.

3 . E = mgh Dot product of gravitational force and vertical displacement is P.E.

4 . Ø = B.A Dot product of area vector and magnetic flux density vector.

5. Angle between the two vectors a and b is a.b/|a| |b| .

RELATED POST

BASICS OF VECTORS PART ONE AND TWO.

Discovery of Neutron,Radius and Density of Nucleus 

Mass Defect and Binding Energy of Nucleus

Radio Activity and Half Life Period

Nuclei Complete Lesson

Vector Concepts part two

This lesson is in continuation with Vectors concepts part one and going through that first will give more convenience to understand the present topic.

c)If three forces (vectors) are to be in equilibrium, then the sum of magnitudes of any two forces must be greater than the magnitude of third force.

d)Lami's theorem:

If a body is in equilibrium under the action of three coplanar forces P,Q,R at angles as shown in the figure,

18. A body of mass 'm' is suspended by a string of length 'l' from a rigid support. It is pulled aside by distance 'x' so that it makes an angle with the vertical by applying a horizontal force F. When the body is in equilibrium.

19. PARALLELOGRAM LAW OF VECTORS (OR FORCES):"If two vectors acting at a point making an angle with each other are represented both in magnitude and direction by the adjacent sides of a parallelogram, then the diagonal drawn from the same point will give the resultant both in magnitude and direction" .

22. POLYGON LAW OF VECTORS :" If number of vectors acting at a point in the same plane in different directions are represented both in magnitude and direction by the adjacent sides of a polygon taken in order, then the closing side taken in the reverse order will give the resultant both in magnitude and direction".

APPLICATIONS OF POLYGON LAW

1. If 'n' equal forces act on a body such that each force makes an angle 2∏ / n with the previous one and the polygon is closed, then the resultant is zero.

If each force of magnitude 'F' makes an angle Θ with previous one, then

a) the resultant is zero, if the number of forces is 2∏/ Θ

b) the resultant is 'F', if the number of forces are 2∏/ Θ - 1
34

Zener diode as Voltage Regulator

Working of Transistor in Common Base Configuration

Transistor in Common Emitter Configuration  

Transistor as Feedback Oscillator

VECTORS CONCEPTS

1.Physical quantities are mainly classified into two types a) Scalars b) Vectors .

2. Scalar quantities re those which have only magnitude.

3. Physical quantities which have both magnitude and direction are called vectors and they should satisfy the parallelogram law of vector addition.

4. Mathematically any directed line segment is called a vector. It has three characteristics.

a) Support (base)
b) Sense (direction)
c) Length (mangnitude or modulus)

5. The magnitude of a vector is a scalar.

6. Electric current, velocity of light have both magnitude and direction but they do not obey the laws of vector addition. Hence they are scalars.

DIFFERENT TYPES OF VECTORS

7. EQUAL VECTORS: Two vectors are said to be equal when their magnitude and direction are equal.

8. NEGATIVE VECTOR: Negative vectors are those which are equal in magnitude but opposite in direction.

9. NULL VECTOR (ZERO VECTOR): It is a vector whose magnitude zero and direction is unspecified.

Examples :

a) Displacement after one complete revolution.

b) Velocity of vertically projected body at the highest point.

10. UNIT VECTOR : It is a vector whose magnitude is unity. A unit vector parallel to a given vector R is given by R ˆr = R

11. REAL VECTOR OR POLAR VECTOR : If the direction of a vector is independent of the coordinate system, then it is called a polar vector.

Example : linear velocity, linear momentum, force, etc.

12. PSEUDO VECTOR: Vectors associated with rotation about an axis and whose direction is changed when the co-ordinate system is changed from left to right, are called axial vectors (or) pseudo vectors.

Example : Torque, Angular momentum, Angular velocity, etc.

13. POSITION VECTOR: It is a vector that represents the position of a particle with respect to the origin of a co-ordinate system. The Position Vection of a point (x, y, z) is r = x i+yj+zk .

ADDITION OF VECTORS

14. There are three laws of addition of vectors.

a) Commutative law : A + B = B + A

b) Associative law : A + (B+C) = (A + B) + C

c) Distributive law : m(A + B) = mA + mB where m is a scalar

RESULTANT OF NUMBER OF VECTORS

15. Resultant is a single vector that gives the total effect of number of vectors.

16. Resultant can be found by using a) Triangle law of vectors b) Parallelogram law of vectors c) Polygon law of vectors .

17. TRIANGLE LAW OF VECTORS: If two given vectors are represented both in magnitude and direction by the two adjacent sides of a triangle, then closing side (third side) taken in the reverse order will give the resultant both in magnitude and direction.

APPLICATIONS OF TRIANGLE LAW :

a) MOTION OF A BOAT CROSSING THE RIVER IN SHORTEST TIME :

If velocities of boat and river are represented with B and R subscripts with V then to cross the river in shortest time, the boat is to be rowed across the river i.e., along normal to the banks of the river.






MOTION OF A BOAT CROSSING THE RIVER IN SHORTEST DISTANCE :


The previous post deals with UNITS AND DIMENSIONS OF PHYSICS PART TWO AND ONE.

Zener diode as Voltage Regulator

Working of Transistor in Common Base Configuration

Transistor in Common Emitter Configuration  

Transistor as Feedback Oscillator