Notes on vectors

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Physics Investigatory Project

2014-15

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




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3. 
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Vectors

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.

Vector Quantities
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:-�⃗ ,??????⃗ etc

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

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

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 3
rd side of triangle taken in opposite
order gives the resultant of two vectors.
Example:-
??????⃗⃗ = �⃗ + �⃗⃗
??????⃗⃗ �⃗⃗


�⃗
Note:-

Same order of Vectors- Head of one vector matches with tail
of other vector.
Example:- �⃗ �⃗⃗

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:- �⃗ �⃗⃗



Parallelogram Law

If 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:-

�⃗ �⃗

9. ??????⃗⃗ �⃗⃗


�⃗
??????⃗⃗=�⃗+�⃗⃗+�⃗+�⃗
 Analytical Method

B Let ø is angle b/w �⃗ & �⃗⃗ and
�⃗ ??????⃗⃗ �⃗ let |�|⃗⃗⃗⃗⃗⃗
= a , |�|⃗⃗⃗⃗⃗⃗ = b and |??????|⃗⃗⃗⃗⃗⃗ =
R


ø ø A
O �⃗⃗ C

From vertex B drop a on OA(extended)
so , cosø = CD/BC & sinø = BA/BC
CD = BC cosø & BA = BCsinø

so , R
2 = (BA)
2 + (OA)
2
R
2 = b
2 sin
2
ø +(OC +CA)
2
R
2 = b
2 sin
2
ø + a
2+b
2 cos
2
ø + 2abcosø
R
= √�
2
+�
2
+2�����ø

Let ??????⃗⃗ makes an angle α with �⃗⃗

10. SUbtraction of vectors

Negative Vector

Negative vector of a given vector is a vector of same magnitude in
opposite direction.
�⃗


-�⃗

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

�⃗ −�⃗⃗ = �⃗ + (−�⃗⃗ )
Graphical Law

11. Example:-
??????⃗⃗ = �⃗ - �⃗⃗
�⃗⃗



�⃗
−�⃗⃗
??????⃗⃗


 Analytical Method

B Let ø is angle b/w �⃗ & �⃗⃗ and
�⃗ ??????⃗⃗ �⃗ let |�|⃗⃗⃗⃗⃗⃗
= a , |−�|⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ = b and |??????|⃗⃗⃗⃗⃗⃗ =
R


ø ø A
O - �⃗⃗ C

From vertex B drop a on OA(extended)
so , cosø = CD/BC & sinø = BA/BC
CD = BC cosø & BA = BCsinø

so , R
2 = (BA)
2 + (OA)
2
R
2 = b
2 sin
2
ø +(OC +CA)
2
R
2 = b
2 sin
2
ø + a
2+b
2 cos
2
ø - 2abcosø
R
= √�
2
+�
2
−2�����ø

Let ??????⃗⃗ makes an angle α with �⃗⃗

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.


�⃗ �⃗ �⃗⃗


�⃗

Vector �⃗ is resolved to �⃗ and �⃗⃗
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 ????????????⃗⃗⃗⃗⃗⃗ vector equal to ??????⃗ and it makes angle ø with X-
axis .Project ??????⃗ along X & Y axis. Let rectangular components of
??????⃗ be Ax and Ay respectively.

14. Y-axis
??????⃗
Ax = Acos ø
Asin ø Ay= Asin ø
Tan ø = Ay/Ax
ø Ax
2
+Ay
2
=A
2
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 ????????????⃗⃗⃗⃗⃗⃗ when projected along space making α,β
(α+β≠90) & γangles with X,Y & Z axis respectively.

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

Thus ax=a cos� , ay=a cos� , az= a cos�

15. Y-axis �⃗



�

� �
X-axis


Z-axis
Further
ax
2+ay
2+ az
2=a
2
so cos�
2
+ cos�
2 + cos�
2 =1



unit vector

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

�̂=
??????⃗⃗
|??????⃗⃗|

16. Orthogonal Unit Vectors

Three unit vectors(called orthogonal unit vectors) �̂,�̂ & �̂ are
used to indicate X,Y & Z axis respectively.



�̂


�̂ �̂



Multiplycation of Vectors


1. Dot(or Scalar ) Product :- �⃗. �⃗⃗

17. 2.Cross (or Vector) Product:- �⃗x �⃗⃗















Dot Product of two vector

Let the two vectors be �⃗& �⃗⃗.

�⃗. �⃗⃗=abcos� where � �� �ℎ� ��??????�� �/� �⃗& �⃗⃗.

18. Ex- W=??????⃗. �⃗ P=??????⃗. �⃗

Dot product of vectors given in Cartesian form

�⃗ = ax �̂+ay �̂+az �̂

�⃗⃗ = bx �̂+by �̂+bz �̂

So
�⃗. �⃗⃗ = ax bx +ay by+ az bz

Note:-(�̂.�̂=�∗�cos0 �̂.�̂=�∗�cos0 �̂.�̂=�∗�cos0
�̂.�̂=�∗�cos90 �̂.�̂=�∗�cos90 �̂.�̂=�∗�cos90)
Properties of Dot Product

i) Commutative:- �⃗. �⃗⃗=�⃗⃗. �⃗
ii) Distributive:- �⃗.( �⃗⃗+ �)⃗⃗⃗⃗ =�⃗. �⃗⃗+�⃗. �⃗
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’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 �⃗& �⃗⃗. �⃗ be the cross product of �⃗��⃗⃗.

|�⃗��⃗⃗|=|�⃗|=absin� where � �� �ℎ� ��??????�� �/� �⃗& �⃗⃗.













Cross product of vectors given in Cartesian form


�⃗ = ax �̂+ay �̂+az �̂
�⃗⃗ = bx �̂+by �̂+bz �̂

20. So
�⃗� �⃗⃗ = (aybz- az by) �̂ +(az bx-axbz)�̂
+ (ax by –ay bx)�̂


Note:-( �̂��̂=�̂ �̂��̂=−�̂
�̂ ��̂=�̂ �̂ � �̂=−�̂
�̂ ��̂=�̂ �̂ ��̂=−�̂
�̂ ��̂=0 �̂ ��̂=0 �̂ ��̂=0 )















Bibliography

21.  WWW.GOOGLE.com
 www.wikipedia.com
 www.ncert.nic.in/ncerts/textbook/textbook.htm