VECTOR

                                                      VECTOR

FOR O LEVEL NOTES

 
CONTENTS

SCALAR AND VECTOR QUANTITIES

VECTOR DIAGRAM

TRIANGLE LAW OF FORCES

PARALLELOGRAM LAW OF FORCES

RELATIVE MOTION

RESOLUTION OF VECTOR


SCALAR AND VECTOR QUANTITY
In the 19 century,mathematicians developed a more suitable way of describing quantities .They divided quantities into two types which are scalar and vector quantities .

SCALAR QUANTITY:Is the physical quantity that has magnitude only but not direction.
A magnitude is a number that shows the  value of the quantity,e.g two meter ,ten kilogram and others.
Example of scalar quantities are Time,Distance,Temperature, Energy, Speed and others.

VECTOR QUANTITY:Is the physical quantity that have both magnitude and direction

Example of vector quantities are Force ,Momentum ,Velocity,Acceleration ad others.

Characteristics of Vectors

The characteristics of vectors are as followed –

• They possess both magnitudes as well as direction
• These change if either the magnitude or direction change or both change

.Comparison between Scalars and Vectors

Criteria	Scalar	Vector
Definition	A scalar is a quantity with magnitude only.	A vector is a quantity with the magnitude as well as direction.
Direction	No direction	Yes there is the direction
Specified by	A number (Magnitude) and a Unit	A number (magnitude), direction and a unit.
Represented by	Quantity symbol	Quantity symbol in bold or an arrow sign above
Example	Mass and Temperature	Velocity and Acceleration

                          

                                         VECTOR DIAGRAM

Vector diagram is the diagram that depict the direction and relative magnitude of a vector.

Vector quantities  can be represented using vector diagrams .

An arrow that has been drawn to scale  and facing a specific direction is  used to represent vector.

The arrow has a head and tail.The length of the arrow  represents  the magnitude of the vector while the arrowhead points in the direction of the vector.

A vector quantity can be represented on paper by a direct line segment.

1. The length of the line segment represents the magnitude of a vector.
2. The arrow head at the end represent the direction of a vector

EQUAL VECTOR

Two vectors are equal if they have the same magnitude  and are the same direction ,even if they do not start from the same point .For example in figure below .

vector a is equal to vector b ,i.e a =b.

  If vector b has the same magnitude as vector a but is opposite in direction , then vector b takes on a negative sign i.e . a =b. It is also correct to say b = -a 

                         VECTOR ADDITION

It is possible to get the sum of two or more vectors .The vector after the addition of two or more vector is called resultant.The resultant vector is equal to the net effect of the vectors  under consideration.

NOTE: Like vectors can be added up . This means that it cannot add unlike vectors such as displacement and acceleration.The resultant vector will always  have the same magnitude and direction  irrespective  of the ordered followed when adding up the vectors

Methods of Vector Addition

There are two methods that are used to sum up two vectors:

1.      Triangle method

2.      Parallelogram method.

Triangle Method

A step-by-step method for applying the head-to-tail method to determine the sum of two or more vectors is given below.

1.Choose a scale and indicate it on a sheet of paper. The best choice of scale is one that will result in a diagram that is as large as possible, yet fits on the sheet of paper.

2.Pick a starting location and draw the first vector to scale in the indicated direction. Label the magnitude and direction of the scale on the diagram (e.g., SCALE: 1 cm = 20 m).

3.Starting from where the head of the first vector ends, draw the second vector to scale in the indicated direction. Label the magnitude and direction of this vector on the diagram.

4.Draw the resultant from the tail of the first vector to the head of the last vector. Label this vector as Resultant or simply R.

5. Using a ruler, measure the length of the resultant and determine its magnitude by converting to real units using the scale (4.4 cm x 20 m/1 cm = 88 m).

6. Measure the direction of the resultant using the counterclockwise convention.

Resultant vector: This is the vector drawn from the starting point of the first vector to the end point of the second vector which is the sum of two vectors

Where:

·         V₁ - First vector

·         V₂ - Second vector

·         R - Resultant vector

EXAMPLE 1

Suppose a man walks starting from point A, a distance of 20m due North, and then 15m due East. Find his new position from A.

Solution

Use scale

1CM Represents 5m

Thus

20m due to North Indicates 4 cm

15m due to East Indicates 3cm.

Demonstration

he position of D is represented by Vector AD of magnitude 25M or 5CM at angle of 36⁰51”

Since

• Tan Q = (Opposite /Adjacent)
• Tan Q = 3cm /4cm
• Q = Tan ^-1 (3/4)
• Q = Tan ^-1(0.75)
• Q = 35º51”

The Resultant displacement is 25m ad direction Q = 36º51”

The Triangle and Parallelogram Laws of Forces

                  TRIANGLE LAW OF FORCES

Triangle Law of Forces states that “If three forces are in equilibrium and two of the forces are represented in magnitude and direction by two sides of a triangle, then the third side of the triangle represents the third force called resultant force.”

EXAMPLE 2

A block is pulled by a force of 4 N acting North wards and another force 3N acting North-East. Find resultant of these two forces.

Demonstration

Scale

1cm Represents 1N

Draw a line AB of 4cm to the North. Then, starting from B, the top vectorofAB, draw a line BC of 3 CM at 45^oEast of North.

Join the line AC and measure the length (AC = 6.5 cm) which represents 6.5N. Hence, AC is the Resultant force of two forces 3N and 4N.

                    PARALLELOGRAM METHOD

In this method, the two Vectors are drawn (usually to scale) with a common starting point , If the lines representing the two vectors are made to be sides of a parallelogram, then the sum of the two vectors will be the diagonal of the parallelogram starting from the common point.
                      PARALLELOGRAM LAW

The Parallelogram Law states that “If two vectors are represented by the two sides given and the inclined angle between them, then the resultant of the two vectors will be represented by the diagonal from their common point of parallelogram formed by the two vectors .

EXAMPLE 3

Two forces AB and AD of magnitude 40N and 60N respectively, are pulling a body on a horizontal table. If the two forces make an angle of 30^o between them find the resultant force on the body.

Solutiuon

Choose a scale.

1cm represents 10N

Draw a line AB of 4cm

Draw a line AD of 6cm.

Make an angle of 30^o between AB and AD. Complete the parallelogram ABCD using the two sides AB and include angle 30^O.

Draw the lineAC with a length of  9.7cm, which is equivalent to 97 N.

The lineAC of the parallelogram ABCD represents the resultant force of AB and AD in magnitude and direction.

EXAMPLE 3

Two ropes, one 3m long and the other and 6m long, are tied to the ceiling and their free ends are pulled by a force of 100N. Find the tension in each rope if they make an angle of 30^o between them.

Solution

1cm represents 1N

Thus

3cm = represent 3m

6cm = represents 6m

Demonstration.

Tension, determined by parallelogram method, the length of diagonal using scale is 8.7 cm, which represents 100N force.

Thus.

Tension in 3m rope = 3 X 100 / 8.7 = 34.5N

Tension in 6m rope = 6 x 100 / 8.7 =69N

Tension force in 3m rope is 34.5N and in 6m rope is 69N

                           RELATIVE MOTION

The Concept of Relative Motion

Relative motion is the motion of the body relative to the moving observer.

The Relative Velocity of two Bodies

Relative velocity (Vr) is the velocity relative to the moving observer.

CASE 1: If a bus in overtaking another, a passenger (observer) in the slower bus sees the Relative Motion

overtaking bus as moving with a very small velocity.

CASE 2: If the passenger was in a stationary bus, then the velocity of the overtaking bus would appear to be greater.

CASE 3: If the observer is not stationary, then to find the velocity of a body B relative to body A add velocity of B to A

EXAMPLE 4

If velocity of body B is V_B and that of body A is V_A, then the velocity of B with respect to A , the relative velocity V_BA is Given by:

V_BA = V_B + (-V_A)

That is

V_BA = V_B – V_A

NOTE:The relative velocity can be obtained Graphically by applying the Triangle or parallelogram method.

For same direction

Vr_BA = V_B - (+V_A)

Vr_BA= V_B – V_A ___________________ (I)

For different direction

Vr_BA = V_B – (-V_A)

Vr_BA = V_B + V_A _______________________ (II)

EXAMPLE 5

A man is swimming at 20 m/s across a river which is flowing at 10 m/s. Find the resultant velocity of the man and his course if the man attempted to swim perpendicular to the water current.

Solution

Scale

1cm represents 2m/s

• The length of AC is 11.25 cm which is 22.5 m/s making an angle of 65º25’ with the water current.
• The diagonal AC represent (in magnitude and direction) the resultant velocity of the man.

                THE CONCEPT OF A COMPONENT OF VECTOR

Components of a vector are vectors which when compounded or added, provides a single vector known as compounded vector.

Resolved vector is a single vector which can be split up into component vectors.

Component vectors are vectors obtained after splitting up or dividing a single vector 

Resolution of a Vector into two Perpendicular Components

Components of a vector are divided into two parts:

1. Horizontal component
2. Vertical component

Find the horizontal and vertical components of a force of 10N acting at 30⁰ to the vertical.

EXAMPLE 6

Find the horizontal and vertical component of the force in the below diagram

MORE QUESTIONS

1.    The air craft P and Q are flying with the same speed. The direction along which P is flying its right angle to the direction along which Q is flying. Find the magnitude of the velocity of the aircraft P relative to the aircraft
2.     An air craft head north east at 320km/h relative to the wind. The wind velocity is 80km/h from the  south. Find the velocity of the aircraft relative to the ground
3.       Acar is travlling due to north at 45km/h. It turns and then travels due to east at 72km/h. Find the magnitude and direction of the resultant velocity of the car .