Notes on vectors

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ghjklzxcvbnmqwertyuiopasdfghjklzxcvbnmqwerty[uioSubmitted by

xcxcvblzxcvbnmqwerShivam Rathi

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Physics Investigatory Project

2014-15

I'd like to express my greatest gratitude to the people who

have helped & supported me throughout my project. I’ m grateful to my Physics Teacher Mr. Chhotelal Gupta for his continuous support for the project, from initial

advice & encouragement to this day. Special thanks of mine goes to my colleagues who helped me in

completing the project by giving interesting ideas, thoughts & made this project easy and accurate.

____________Shivam Rathi

Vectors

Content

1. Introduction

2. Representation of Vectors

3. Addition and Subtraction of Vectors

4. Resolution of vector ( i ) Rectangular Component (ii) 3-D resolution of vector

5. Unit Vector

6. Multiplication of Vector ( i ) Dot Product (ii) Cross Product

Introduction

Scalar QuantitiesPhysical quantities having magnitude alone are known as Scalar quantities. Examples:- Mass, Time, Distance etc.

Vector QuantitiesPhysical quantities having both magnitude and direction and also follow vector rule of addition are known as vector quantities. Examples:- Displacement, Momentum ,Force etc.

Tensor QuantitiesPhysical Quantities which are neither vectors nor scalars are known as tensor quantities. Examples :- Moment of inertia, Stress, etc.

Note:- Some quantities like area, length, angular velocity, etc. are treated as both scalars as well as vectors.

Representation of a vector

Vectors are represented by alphabets (both small and capital) with an arrow at its top. Examples:-a , A etc

Magnitude of vector is represented as a or |a|.

Graphically a vector is represented as an arrow, and head indicating direction of vector. Example :- head(indicating direction) a tail of vector

Addition of vectorsGraphical LawAccording to this law if two vectors are represented in magnitude and direction by two consecutive sides of a triangle taken in same order then the 3rd side of triangle taken in opposite order gives the resultant of two vectors.Example:- R = a + b R b

a Note:- Same order of Vectors- Head of one vector matches with tail of other vector.

Example:- a b Opposite order of Vectors- Two vectors are said to be in opposite order if either tail matches with tail or head matches with head of other vector.Example:- a b

Parallelogram LawIf two vectors are represented in magnitude and direction by two adjacent side of parallelogram intersecting at point then the resultant is obtained by the diagonal of the parallelogram passing through the same point.

Polygon Law

It states that if a no. of vectors are represented in magnitude and direction by sides of a polygon taken in same order then the resultant is obtained by closing side of polygon taken in opposite order.Example:- d c

R b

a R=a+b+c+d Analytical Method B Let ø is angle b/w a & b and a R a let |a| = a , |b| = b and |R| = R ø ø A O b C

From vertex B drop a on OA(extended) so , cos ø = CD/BC & sin ø = BA/BC CD = BC cos ø & BA = BCsin ø so , R2 = (BA)2 + (OA)2

R2 = b2 sin2 ø +(OC +CA)2 R2 = b2 sin2 ø + a2+b2 cos2ø + 2abcos ø R = √a2+b2+2abcos ø

Let R makes an angle α with b

SUbtraction of vectorsNegative VectorNegative vector of a given vector is a vector of same magnitude in opposite direction. a

-a

Subtraction of b from a is nothing but addition of a +(−b ) .

a −b = a + (−b )Graphical LawExample:- R = a - b b

a −b R

Analytical Method B Let ø is angle b/w a & b and a R a let |a| = a , |−b| = b and |R| = R ø ø A O - b C

From vertex B drop a on OA(extended)

so , cos ø = CD/BC & sin ø = BA/BC CD = BC cos ø & BA = BCsin ø so , R2 = (BA)2 + (OA)2

R2 = b2 sin2 ø +(OC +CA)2 R2 = b2 sin2 ø + a2+b2 cos2ø - 2abcos ø R = √a2+b2−2abcos ø

Let R makes an angle α with b

Resolution Of VectorsThe process of splitting a vector into two or more vectors along different directions is called “resolution of vectors”.

The splitted vectors are called “components of given vector”.

A vector can have ‘infinite’ components.

Resolution of vectors is reverse of addition of vector.

c c b a

Vector c is resolved to a and b Rectangular component

If the components of a vector are mutually perpendicular ,the components are called rectangular components of the given vector.

Resolution in 2-Dimensions

Consider OA vector equal to A and it makes angle ø with X- axis .Project A along X & Y axis. Let rectangular components of A be Ax and Ay respectively.

Y-axis A Ax = Acos ø Asin ø Ay= Asin ø Tan ø = Ay/Ax

ø Ax2+Ay2=A2

O Acos ø X-axis

(A vector can have maximum 2 rectangular component in a plane & maximum 3 in space)

3-d Resolution of vectorConsider a vector OA when projected along space making α,β (α+β≠90) & γangles with X,Y & Z axis respectively.

Let OA = a &|OA| =a.Let the rectangular components of a be ax ,ay& az.

Thus ax=acos α , ay=a cos β , az= acos γ

Y-axis a β

γ α X-axis

Z-axisFurther ax2+ay2+ az2=a2

so cos α 2+ cos β 2 +cos γ2 =1

unit vectorVector having magnitude as unity are called unit vector . They are represented as a (‘cap’or ‘hat’).They are used to indicated direction .A unit vector may also be defined as vector divide by its magnitude i.e.

a= a|a|

Orthogonal Unit Vectors

Three unit vectors(called orthogonal unit vectors) i , j∧k are used to indicate X,Y & Z axis respectively. j

k i

Multiplycation of Vectors

1. Dot(or Scalar ) Product :- a.b 2.Cross (or Vector) Product:- axb

Dot Product of two vectorLet the two vectors be a&b.

a.b=abcos α where α is the angle b/w a&b.

Ex- W=F.s P=F.v

Dot product of vectors given in Cartesian form

a = ax i+ay j+azk

b = bx i+by j+bzk

So a.b = ax bx +ay by+ az bz

Note:-(i . i=i∗icos0 j . j= j∗ jcos 0 k . k=k∗k cos0 i . j=i∗ jcos 90 i .k=i∗kcos 90 k . j= j∗k cos90)Properties of Dot Product

i) Commutative:- a.b=b.aii) Distributive:- a.(b+ c¿¿ =a.b+a.c

Cross Product “Cross -Product” of two vectors is another vector where magnitude is equal to the product of the magnitude of the vectors & sin of the smaller angle b/w them.The dir’n of this vector is perpendicular to the plane containing the given vectors & given by Right Hand Thumb Rule or Screw Rule.

Let the two vectors be a&b.c be the cross product of a x b.|a x b|=|c|=absinα where α is the angle b/w a&b.

Cross product of vectors given in Cartesian form

a = ax i+ay j+azk b = bx i+by j+bzk

So a x b = (aybz- az by) i +(az bx-axbz¿ j+ (ax by –ay bx)k

Note:-(i x j= k j x i=−k

j x k=i k x j=−i k x i= j i x k=− j i x i=0 j x j=0 k x k=0)

Bibliography WWW.GOOGLE.com www.wikipedia.com www.ncert.nic.in/ncerts/textbook/textbook.htm