2. 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
4. 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
5. Introduction
Scalar Quantities
Physical quantities having magnitude alone are known as
Scalar quantities.
Examples:- Mass, Time, Distance etc.
VectorQuantities
Physical quantities having both magnitude and direction
and also follow vector rule of addition are known as vector
quantities.
Examples:- Displacement, Momentum ,Force etc.
Tensor Quantities
Physical Quantities which are neither vectors nor scalars
are known as tensor quantities.
Examples :- Moment of inertia, Stress, etc.
6. 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:-𝑎⃗ ,𝐴⃗ etc
Magnitude of vector is represented as a or | 𝑎⃗|.
Graphically a vector is represented as an arrow, and
head indicating direction of vector.
Example :-
head(indicating direction)
𝑎⃗
tail of vector
7. Addition of vectors
Graphical Law
According 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:-
𝑅⃗⃗ = 𝑎⃗ + 𝑏⃗⃗
𝑅⃗⃗ 𝑏⃗⃗
𝑎⃗
Note:-
Same order of Vectors- Head of one vector matches with tail
of other vector.
Example:- 𝑎⃗ 𝑏⃗⃗
8. 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:- 𝑎⃗ 𝑏⃗⃗
Parallelogram Law
If two vectors are represented in magnitudeand direction by two
adjacent sideof 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
directionby sides of a polygon taken in same order then the resultant is
obtained by closing sideof polygon taken in oppositeorder.
Example:-
𝑑⃗ 𝑐⃗
9. 𝑅⃗⃗ 𝑏⃗⃗
𝑎⃗
𝑅⃗⃗=𝑎⃗+𝑏⃗⃗+𝑐⃗+𝑑⃗
Analytical Method
B Let ø is angle b/w 𝑎⃗ & 𝑏⃗⃗ and
𝑎⃗ 𝑅⃗⃗ 𝑎⃗ let | 𝑎|⃗⃗⃗⃗⃗⃗ = a , | 𝑏|⃗⃗⃗⃗⃗⃗ = b and | 𝑅|⃗⃗⃗⃗⃗⃗ =
R
ø ø A
O 𝑏⃗⃗ 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 cos2
ø + 2abcos ø
R = √ 𝑎2 + 𝑏2 + 2𝑎𝑏𝑐𝑜𝑠ø
Let 𝑅⃗⃗ makes an angle α with 𝑏⃗⃗
10. SUbtraction of vectors
Negative Vector
Negativevector of a given vector is a vector of same magnitude in
oppositedirection.
𝑎⃗
- 𝑎⃗
Subtraction of 𝑏⃗ from 𝑎⃗ is nothing but additionof 𝑎⃗ +(− 𝑏⃗⃗ ) .
𝑎⃗ − 𝑏⃗⃗ = 𝑎⃗ + (− 𝑏⃗⃗ )
Graphical Law
11. Example:-
𝑅⃗⃗ = 𝑎⃗ - 𝑏⃗⃗
𝑏⃗⃗
𝑎⃗
− 𝑏⃗⃗
𝑅⃗⃗
Analytical Method
B Let ø is angle b/w 𝑎⃗ & 𝑏⃗⃗ and
𝑎⃗ 𝑅⃗⃗ 𝑎⃗ let | 𝑎|⃗⃗⃗⃗⃗⃗ = a , |−𝑏|⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ = b and | 𝑅|⃗⃗⃗⃗⃗⃗ =
R
ø ø A
O - 𝑏⃗⃗ 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 cos2
ø - 2abcos ø
R = √ 𝑎2 + 𝑏2 − 2𝑎𝑏𝑐𝑜𝑠ø
Let 𝑅⃗⃗ makes an angle α with 𝑏⃗⃗
12. Resolution Of Vectors
The 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.
13. Resolution of vectors is reverse of addition of vector.
𝑐⃗ 𝑐⃗ 𝑏⃗⃗
𝑎⃗
Vector 𝑐⃗ is resolved to 𝑎⃗ and 𝑏⃗⃗
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 𝑂𝐴⃗⃗⃗⃗⃗⃗ vector equal to 𝐴⃗ and it makes angle ø with X-
axis .Project 𝐴⃗ along X & Y axis. Let rectangular components of
𝐴⃗ be Ax and Ay respectively.
14. Y-axis
𝐴⃗
Ax = Acos ø
Asinø Ay= Asin ø
Tan ø = Ay/Ax
ø Ax
2+Ay
2=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 vector
Consider a vector 𝑂𝐴⃗⃗⃗⃗⃗⃗ when projected along space making α,β
(α+β≠90) & γangles with X,Y & Z axis respectively.
Let 𝑂𝐴⃗⃗⃗⃗⃗⃗ = 𝑎⃗ & |𝑂𝐴⃗⃗⃗⃗⃗⃗| =a.
Let the rectangular components of 𝑎⃗ be ax ,ay& az.
Thus ax=a cos 𝛼 , ay=a cos 𝛽 , az= a cos 𝛾
15. Y-axis 𝑎⃗
𝛽
𝛾 𝛼
X-axis
Z-axis
Further
ax
2+ay
2+ az
2=a2
so cos 𝛼 2+ cos 𝛽 2 + cos 𝛾2 =1
unit vector
Vector having magnitude as unity are called unit vector . They
are represented as 𝑎̂ (‘cap’or ‘hat’).They are used to indicated
direction .A unit vector may also be defined as vector divide by
its magnitude i.e.
𝑎̂=
𝑎⃗⃗
| 𝑎⃗⃗|
16. Orthogonal Unit Vectors
Three unit vectors(called orthogonal unit vectors) 𝑖̂, 𝑗̂ & 𝑘̂ are
used to indicate X,Y & Z axis respectively.
𝑗̂
𝑘̂ 𝑖̂
Multiplycation of Vectors
1. Dot(or Scalar ) Product :- 𝑎⃗. 𝑏⃗⃗
17. 2.Cross (or Vector) Product:- 𝑎⃗x 𝑏⃗⃗
Dot Product of twovector
Let the two vectors be 𝑎⃗& 𝑏⃗⃗.
𝑎⃗. 𝑏⃗⃗=abcos 𝛼 where 𝛼 𝑖𝑠 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑏/𝑤 𝑎⃗& 𝑏⃗⃗.
18. Ex- W= 𝐹⃗. 𝑠⃗ P= 𝐹⃗. 𝑣⃗
Dot product of vectors given in Cartesian form
𝑎⃗ = ax 𝑖̂+ay 𝑗̂+az 𝑘̂
𝑏⃗⃗ = bx 𝑖̂+by 𝑗̂+bz 𝑘̂
So
𝑎⃗. 𝑏⃗⃗ = ax bx +ay by+ az bz
Note:-(𝑖̂. 𝑖̂ = 𝑖 ∗ 𝑖 cos 0 𝑗̂. 𝑗̂ = 𝑗 ∗ 𝑗 cos 0 𝑘̂. 𝑘̂ = 𝑘 ∗ 𝑘 cos 0
𝑖̂. 𝑗̂ = 𝑖 ∗ 𝑗 cos 90 𝑖̂. 𝑘̂ = 𝑖 ∗ 𝑘 cos9 0 𝑘̂. 𝑗̂ = 𝑗 ∗ 𝑘 cos 90)
Properties of Dot Product
i) Commutative:- 𝑎⃗. 𝑏⃗⃗= 𝑏⃗⃗. 𝑎⃗
ii) Distributive:- 𝑎⃗.( 𝑏⃗⃗ + 𝑐)⃗⃗⃗⃗ = 𝑎⃗. 𝑏⃗⃗+ 𝑎⃗. 𝑐⃗
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.
19. The dir’nof 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 𝑎⃗& 𝑏⃗⃗. 𝑐⃗ be the cross product of 𝑎⃗ 𝑥𝑏⃗⃗.
|𝑎⃗ 𝑥𝑏⃗⃗|=| 𝑐⃗|=absin 𝛼 where 𝛼 𝑖𝑠 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑏/𝑤 𝑎⃗& 𝑏⃗⃗.
Cross product of vectors given in Cartesian form
𝑎⃗ = ax 𝑖̂+ay 𝑗̂+az 𝑘̂
𝑏⃗⃗ = bx 𝑖̂+by 𝑗̂+bz 𝑘̂
