Physics Presentation

04/12/23 vector analysis 2

Members of the Group

Ammar MaqsoodAmmar Maqsood M Adil BashirM Adil Bashir Hassaan Ahmed UsmaniHassaan Ahmed Usmani Naeem NassirudinNaeem Nassirudin


Scalars Vectors A scalar quantity is one

which can be described fully by just stating its magnitude.

Some examples are Mass time length temperature density speed energy and volume

A vector quantity is one which can only be fully described if its magnitude and direction stated.

Some examples are displacement velocity acceleration force momentum magnetic density and

electric intensity.


Why vectors are important? Vectors are fundamental in the physical

sciences. They can be used to represent any quantity that has both a magnitude and direction, such as velocity, the magnitude of which is speed. For example, the velocity 5 meters per second upward could be represented by the vector (0,5). Another quantity represented by a vector is force, since it has a magnitude and direction.


Properties of a Vector

A vector is a quantity that has both direction and magnitude. Let a vector be denoted by the symbol A.

The magnitude of A is |A| ≡ A We can represent vectors as

geometric objects using arrows. The length of the arrow corresponds to the magnitude of the vector. The arrow points in the direction of the vector

O

A

|A| ≡ A


Types of Vector Null Vector

Vector with zero magnitude Position Vector

Vector starting from origin Free Vector

Vector starting from anywhere but origin Unit Vector

Vector with magnitude of 1 Equal Vectors

Two vectors equal in magnitude and direction Opposite Vectors

Two vectors equal in magnitude but opposite in direction.


Laws of vector algebra

A+B=B+A (Commutative law of addition)

A+(B+C)=(A+B)+C (Associative law of addition)

mA=Am (Commutative law of Multiplication)

m(nA)=(mn)A (Associative law of Multiplication)

(m+n)A=mA+nA (Distributive law)

m(A+B)=mA+mB (Distributive law)


Addition of Vectors

The addition of two vectors yields another vector known as Resultant vector.

For example if vector A and vector B are added their sum will be equal to (A+B).


Methods of Addition of Vectors

There are 3 methods for addition of Vectors.

1. Addition Algebraically1. For vectors precisely along X or Y axis.

2. Parallelogram law of addition

3. Triangle law of addition.


Parallelogram law Vector P and Q are

drawn from same origin.

Straight lines are drawn parallel to both vectors so as to form a parallelogram.

The resultant (P+Q) is represented by the diagonal of the parallelogram that passes through the origin.

P Q+

P

PQ Q+


Triangle law (head to tail rule) The result of two

vectors could be determined by drawing a triangle.

Vector Q and P are drawn in such a way that the tail of vector Q touches head of vector P.

The resultant (P+Q) is represented by the third side of triangle from tail of P to head of Q.

P Q+

P

QPQ+


Subtraction of vectors The subtraction of two

vectors can be treated as the addition of a negative vector.

(P-Q)=P+(-Q)

The vector (P-Q) can then be determined by any of the two methods.

P Q-

P

-QP-Q

P

P-Q -Q

=

OR


Resolution of a vector

Vector R could be considered to be the resultant of two vectors.

R=A+B

Here the vectors A and B are known as the components of vectors.


Resolution of a vector It is useful to find the

components of a vector R in two mutually perpendicular directions. This process is known as resolving a vector into components.

The magnitude of the two components can be written in the form Rcos and Rsin

R

0Rcos0

Rsi

n0

R

Rcos

Rsin


Triangle of forces If three forces acting on a point can

be represented in magnitude and direction by three sides of a triangle taken in order, then the three forces are in equilibrium.

The converse is also true: Three forces acting on a point are in

equilibrium, they can be represented in magnitude and direction by sides of a triangle taken in order.


Triangle of forces

By the triangle of vectors, the resultant of P and Q is represented in magnitude and direction by side of OC of the triangle OAC.If third force R is equal in magnitude to (P+Q) but in opposite direction, then the point O is in equilibrium and also R could be represented by the side CO of the triangle.

o

P

Q

R

A C

OR

P

Q

P+Q

A C

O

P R

Q


Polygon of Forces

When more than three coplanar forces act on a point , the resultant (or vector sum) of forces can be found by drawing a polygon of forces.


Polygon of Forces If forces acting on

a point can be represented in magnitude and direction by sides of a polygon taken in order ,then the forces are in equilibrium.

o

S

T

P

QR

Q

RS

P

T

A B

C

D

E


Resultant of a number of forces If point O is being

acted upon by a number of coplanar forces such as A,B,C,D,E do not form a closed polygon then the forces are not in equilibrium

The resultant in magnitude and direction is represented by R.

A

B

C

D

E

A

B

CDE

R

o


When vector is multiplied by another vector in some cases a scalar quantity is obtained whereas in some other cases a vector quantity obtained .There are two types of product.

Scalar product Vector product

PRODUCT OF VECTORS

M.Adil Bashir


Scalar product or Euclidean inner product

INTRODUCTION


WHY SCALAR PRODUCT IS USE

Geometrically, the scalar product is useful for finding the direction between arbitrary vectors in space.

involve the components of the two vectors and since the magnitudes A and B can be calculated from the components using:

A . B = A B COSθ


SPECIAL CASES Scalar product is maximum

Angle b/w vectors is zero (θ = 0°) Vectors are in same direction

Scalar product is minimum Angle b/w vectors is right (θ = 90°) Vectors are perpendicular to each

other Scalar product is negative

Angle b/w vectors is 180 (θ = 180°) Vectors are opposite in direction

X

Y

X

X

Y

Y


SCALAR PRODUCT LENGTH/MAGNITUDE OF A VECTOR

The Dot Product of a vector with itself is always equal to its magnitude squared

PARALLEL VECTORS

PERPENDICULAR VECTORS

When A and B are parallel to each other, their Dot Product is identical to the ordinary multiplication of their sizes

When A and B are perpendicular to each other, their Dot Product is always Zero


SCALAR PRODUCT

Since i and j and k are all one unit in length and they are all mutually perpendicular, we have

i. i = j. j = k. k = 1 and i. j = j. i = i. k = k. i = j. k = k. j = 0.


SCALAR PRODUCT IN COMPONENT FORM

Two vectors in component forms are written as a=axi+ayj+azk b=bxi+byj+bzk In evaluating the product, we make use of the

fact that multiplication of the same unit vectors is 1, while multiplication of different unit vectors is zero. The dot product evaluates to scalar terms as :

a.b=(axi+ayj+azk).(bxi+byj+bzk) ⇒a.b=axi.bxi+ayj.byj+azk.bzk ⇒a.b=axbx+ayby+azbz


LAWS OF DOT PRODUCT

Commutative law

A . B = B . A

Distributive law

A . ( B + C ) = A . B + A . C

B

BCOSθ

AB

ACOSθ

A

B

A

R=A+BC


EXAMPLES OF SCALAR PRODUCT

WORK DONE Definition

Example:

The man is pulling the block with a constant force a so that it moves along the horizontal ground . The work done in moving the block through a distance b is then given by the distance moved through multiplied by the magnitude of the component of the force in the direction of motion.



POWER Definition

P = F . V

ELECTRIC FLUX

Definition

Electric flux = E . A


A reader has used the dot product to help him analyze the movements of a tagged right whale. He had the x and y coordinates for a set of positions of the whale and wanted to calculate the angles turned through between successive sections of its journey. He found each section as a vector by calculating the differences between the pairs of x and y coordinates of the endpoints. Then he used the dot product of each successive pair of vectors to find the angle between those two legs of the whale's journey.


A right whale fluke © the New England Aquarium


CROSS PRODUCT

Q.What is Cross Product ?

Hassaan Ahmed Usmani


What is Cross Product

In mathematics, the Cross Product is a binary operation on two vectors in three-dimensional space that results in another vector which is perpendicular to the two input vectors.

However the Dot Product produces a scalar result.


Cross Product The cross product of two vectors a and

b is denoted by a × b. The cross product is given by the

formula

Where θ is the measure of the angle between a and b, a and b are the magnitudes of vectors a and b, and is a unit vector perpendicular to the plane containing a and b.


The right hand rule

By using the right hand rule we can find out the direction of the vector.


Cross Product of standard basic vectors


Geometric Meaning

The magnitude of the cross product can be interpreted as the unsigned area of the parallelogram having a and b as sides


Examples of cross product

AreaThe magnitude of the cross

product a b is the area of the parallelogram with sides a and b.


Examples of cross product

Volume :To find the volume of a

paralleliped with sides a, b, c:

we get


Torque When force is applied to a lever fixed to a point, some

of the force goes towards rotation while the rest goes towards stretching the lever.

The magnitude of the torque is also proportional to the length of the lever, and has a direction depending on which direction the lever pivots.


POINT,LINE AND PLANE

By Naeem Nassiruddin


What is Line?

A line is a series of points that extends without end in two directions.

A line is made up of an infinite number of points.

The line below is named line AB or line BA.

A

B


Line

Three points may lie on the same line. These points are Collinear.

Points that DON’T lie on the same line are Not collinear.

R

T

S

U

V


Line

Horizontal lines have a slope of zero.

Vertical lines are said to have infinite slope.


Skew Line

Skew lines only happen in space. They are non coplanar lines that never intersect. Unlike parallel lines, however, they don't always have a set distance between them, nor do they always have the same direction.

Two lines are skew if they are not both contained in a single plane.


Postulates of Geometry Postulate 1

Two points determine a unique line.

Postulate 2 If two distinct lines intersect, then their intersection is

a point.

Postulate 3 Three no collinear points determine a unique plane.

Q

P

Tl

m

A

BC


Postulate 4 If two distinct planes intersect, then their

intersection is a line.

M

N

D E


Equation of line in 3D


What is Plane? A plane is a flat surface that extends

without end in all directions. Points that lie in the same plane are

coplanar. Points that do not lie in the same plane

are non coplanar.

B

A

C


y

x

5-4 -2 1 3 5

5

-4

-2

1

3

5

-5 -1 4

-5

-1

4

-3

-5

2

2-5

-3

Quadrant I(+, +)

Quadrant II

(–, +)

Quadrant III

(–, –)

Quadrant IV

(+, –)

Quadrants of Plane


Equation of Plane


Equation of plane in 3D

This time, the locus is a plane.


Distance b/w point to plane


Thank You !!Thank You !!

Any Questions??????

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