Vector Analysis | Mathematics Optional Notes for UPSC PDF Download

 Page 1


 
 
 
VECTOR ANALYSIS
 
 
? Vector analysis is a handy way to write and show math relationships between 
things we measure in the physical world. There are two types of things we look 
at: scalars and vectors.  
 
? Scalars are fully described when we know their size or amount. Examples 
include mass or temperature. 
 
? Vectors need both size and direction to be fully described. Think of things like 
displacement, velocity, or force. 
 
? We write vectors with bold letters or a line above the symbol. When drawn, a 
vector is shown as an arrow. The length of the arrow tells us its size, the direction 
it points tells us its direction, and the arrowhead shows its sense or which way it's 
going. 
Vector Algebra 
? If two-line segments are parallel, have the same length, and go in the same 
direction, we call them equal vectors. If they have the same length but go in 
opposite directions, we call one of them the negative of the other. For example, if 
one vector is shown as "v ? ", its opposite would be "-v ?". 
? If we want a vector that's parallel to " ???” but longer by a certain factor "m", we 
write it as "m·??? ". This means it's "m" times as long as "v ? " and has the same 
direction. 
? Let ???
 and ???
 be any two vectors. 
? if we place the initial point of ???
 on the terminal point of ???
  , the vector ???
, drawn 
from the initial point of ???
 to the terminal point of ???
 is defined as the vector sum of 
???
 and ???
. 
 
Page 2


 
 
 
VECTOR ANALYSIS
 
 
? Vector analysis is a handy way to write and show math relationships between 
things we measure in the physical world. There are two types of things we look 
at: scalars and vectors.  
 
? Scalars are fully described when we know their size or amount. Examples 
include mass or temperature. 
 
? Vectors need both size and direction to be fully described. Think of things like 
displacement, velocity, or force. 
 
? We write vectors with bold letters or a line above the symbol. When drawn, a 
vector is shown as an arrow. The length of the arrow tells us its size, the direction 
it points tells us its direction, and the arrowhead shows its sense or which way it's 
going. 
Vector Algebra 
? If two-line segments are parallel, have the same length, and go in the same 
direction, we call them equal vectors. If they have the same length but go in 
opposite directions, we call one of them the negative of the other. For example, if 
one vector is shown as "v ? ", its opposite would be "-v ?". 
? If we want a vector that's parallel to " ???” but longer by a certain factor "m", we 
write it as "m·??? ". This means it's "m" times as long as "v ? " and has the same 
direction. 
? Let ???
 and ???
 be any two vectors. 
? if we place the initial point of ???
 on the terminal point of ???
  , the vector ???
, drawn 
from the initial point of ???
 to the terminal point of ???
 is defined as the vector sum of 
???
 and ???
. 
 
???
=???
+???
 
This is the parallelogram law for the composition of two forces (which are vectors). 
Evidently the same vector ?? will be obtained, if we place the initial point of ???
 on the 
terminal point of ????
 : 
 So ???
=???
+???
=???
+???
.  
So, vector addition is commutative (i.e.) the order of vectors appearing in a sum is 
immaterial. The definition of vector addition can be extended to any number of vectors. 
 
 
The addition is also associative, as can be seen from the above diagrams 
???
=(???
+???
)+???
=???
+(???
+???
) 
A unit vector is a vector that has a length of 1.  
Imagine a right-handed rectangular coordinate system.  
Any vector in three dimensions can be shown with its starting point at the origin, which 
we'll call "O".  
Let's say "?? 1
,?? 2
,?? 3
" are the coordinates of the end point of the vector "A" when it starts 
from "O". Let ???,???
 the unit vector in the ?? ,?? ,?? directions respectively, 
 
Page 3


 
 
 
VECTOR ANALYSIS
 
 
? Vector analysis is a handy way to write and show math relationships between 
things we measure in the physical world. There are two types of things we look 
at: scalars and vectors.  
 
? Scalars are fully described when we know their size or amount. Examples 
include mass or temperature. 
 
? Vectors need both size and direction to be fully described. Think of things like 
displacement, velocity, or force. 
 
? We write vectors with bold letters or a line above the symbol. When drawn, a 
vector is shown as an arrow. The length of the arrow tells us its size, the direction 
it points tells us its direction, and the arrowhead shows its sense or which way it's 
going. 
Vector Algebra 
? If two-line segments are parallel, have the same length, and go in the same 
direction, we call them equal vectors. If they have the same length but go in 
opposite directions, we call one of them the negative of the other. For example, if 
one vector is shown as "v ? ", its opposite would be "-v ?". 
? If we want a vector that's parallel to " ???” but longer by a certain factor "m", we 
write it as "m·??? ". This means it's "m" times as long as "v ? " and has the same 
direction. 
? Let ???
 and ???
 be any two vectors. 
? if we place the initial point of ???
 on the terminal point of ???
  , the vector ???
, drawn 
from the initial point of ???
 to the terminal point of ???
 is defined as the vector sum of 
???
 and ???
. 
 
???
=???
+???
 
This is the parallelogram law for the composition of two forces (which are vectors). 
Evidently the same vector ?? will be obtained, if we place the initial point of ???
 on the 
terminal point of ????
 : 
 So ???
=???
+???
=???
+???
.  
So, vector addition is commutative (i.e.) the order of vectors appearing in a sum is 
immaterial. The definition of vector addition can be extended to any number of vectors. 
 
 
The addition is also associative, as can be seen from the above diagrams 
???
=(???
+???
)+???
=???
+(???
+???
) 
A unit vector is a vector that has a length of 1.  
Imagine a right-handed rectangular coordinate system.  
Any vector in three dimensions can be shown with its starting point at the origin, which 
we'll call "O".  
Let's say "?? 1
,?? 2
,?? 3
" are the coordinates of the end point of the vector "A" when it starts 
from "O". Let ???,???
 the unit vector in the ?? ,?? ,?? directions respectively, 
 
The vectors ?? 1
,?? 2
?? 1
?? 3
?? , are the rectangular component vectors of ???
 in the ?? ,?? and 
?? ???? rections respectively. The sum or resultant of these is the vector ???
 so that we can 
write 
???
 =?? 1
???+?? 2
??¨+?? 3
???
 
The magnitude of 
???
=v?? 1
2
+?? 2
2
+?? 3
2
=|???
| 
A null vector is a vector (denoted by ???
 ) is a vector whose magnitude is zero. 
Vector Multiplication 
There are two types of multiplication called the dot product and the cross product. 
Dot product 
? The dot product or scalar product of two vectors ???
 and ???
 denoted by ???
·???
 (A dot 
B) defined as the product of the magnitudes of ???
  and ???
 and the cosine of the 
angle between them  
???
·???
=???? cos?? , 0<?? =?? 
???
·???
 Is a scalar 
 
The following results readily follow from the above definition  
(1) ???
·???
=???
·???
 
(2) ???
·(???
+???
)=???
·???
+???
·???
 
(3) ???·???=???·???=???
·???
=1; 
???·???=???·????
=???
·???=0 
(4) If ???
=?? 1
???+???
2
??¨+?? 3
???
 and 
????
=?? 1
???+?? 2
???+?? 3
????
, then  
Page 4


 
 
 
VECTOR ANALYSIS
 
 
? Vector analysis is a handy way to write and show math relationships between 
things we measure in the physical world. There are two types of things we look 
at: scalars and vectors.  
 
? Scalars are fully described when we know their size or amount. Examples 
include mass or temperature. 
 
? Vectors need both size and direction to be fully described. Think of things like 
displacement, velocity, or force. 
 
? We write vectors with bold letters or a line above the symbol. When drawn, a 
vector is shown as an arrow. The length of the arrow tells us its size, the direction 
it points tells us its direction, and the arrowhead shows its sense or which way it's 
going. 
Vector Algebra 
? If two-line segments are parallel, have the same length, and go in the same 
direction, we call them equal vectors. If they have the same length but go in 
opposite directions, we call one of them the negative of the other. For example, if 
one vector is shown as "v ? ", its opposite would be "-v ?". 
? If we want a vector that's parallel to " ???” but longer by a certain factor "m", we 
write it as "m·??? ". This means it's "m" times as long as "v ? " and has the same 
direction. 
? Let ???
 and ???
 be any two vectors. 
? if we place the initial point of ???
 on the terminal point of ???
  , the vector ???
, drawn 
from the initial point of ???
 to the terminal point of ???
 is defined as the vector sum of 
???
 and ???
. 
 
???
=???
+???
 
This is the parallelogram law for the composition of two forces (which are vectors). 
Evidently the same vector ?? will be obtained, if we place the initial point of ???
 on the 
terminal point of ????
 : 
 So ???
=???
+???
=???
+???
.  
So, vector addition is commutative (i.e.) the order of vectors appearing in a sum is 
immaterial. The definition of vector addition can be extended to any number of vectors. 
 
 
The addition is also associative, as can be seen from the above diagrams 
???
=(???
+???
)+???
=???
+(???
+???
) 
A unit vector is a vector that has a length of 1.  
Imagine a right-handed rectangular coordinate system.  
Any vector in three dimensions can be shown with its starting point at the origin, which 
we'll call "O".  
Let's say "?? 1
,?? 2
,?? 3
" are the coordinates of the end point of the vector "A" when it starts 
from "O". Let ???,???
 the unit vector in the ?? ,?? ,?? directions respectively, 
 
The vectors ?? 1
,?? 2
?? 1
?? 3
?? , are the rectangular component vectors of ???
 in the ?? ,?? and 
?? ???? rections respectively. The sum or resultant of these is the vector ???
 so that we can 
write 
???
 =?? 1
???+?? 2
??¨+?? 3
???
 
The magnitude of 
???
=v?? 1
2
+?? 2
2
+?? 3
2
=|???
| 
A null vector is a vector (denoted by ???
 ) is a vector whose magnitude is zero. 
Vector Multiplication 
There are two types of multiplication called the dot product and the cross product. 
Dot product 
? The dot product or scalar product of two vectors ???
 and ???
 denoted by ???
·???
 (A dot 
B) defined as the product of the magnitudes of ???
  and ???
 and the cosine of the 
angle between them  
???
·???
=???? cos?? , 0<?? =?? 
???
·???
 Is a scalar 
 
The following results readily follow from the above definition  
(1) ???
·???
=???
·???
 
(2) ???
·(???
+???
)=???
·???
+???
·???
 
(3) ???·???=???·???=???
·???
=1; 
???·???=???·????
=???
·???=0 
(4) If ???
=?? 1
???+???
2
??¨+?? 3
???
 and 
????
=?? 1
???+?? 2
???+?? 3
????
, then  
???
·???
=?? 1
?? 1
+?? 2
?? 2
+?? 3
?? 3
.
???
·???
=?? 1
2
+?? 2
2
+?? 3
 
2
=|???
|
2
???
·???
=?? 1
 
2
+?? 2
2
+?? 3
 
2
=|???
|
2
 
(5) If ???
·????
=0 and ???
 and ????
 are not null vectors, then A and B are perpendicular. 
 
Vector Product 
? The cross or vector product of ???
 and ???
 is a vector ???
=???
×???
 (A cross ?? ).  
? The magnitude of ???
×??~
 is defined as the product of the magnitudes of ???
 and ???
 
and the sine of the angle ?? between them.  
? The direction of the vector ???
=???
×????
 is perpendicular to the plane of ???
 and ???
 and 
such that ???
,???
 and ???
 from a right-handed system. 
???
×???
=???? (sin ?? )???: 0=?? =?? 
? Where ???
 is a unit vector in the direction of ???
×???
. If ???
 is parallel to ???
,??~
×???
=0 
 
The following results are easily deduced: 
(1) ???
×???
=????
¯
×???
 
(2) ???
×(???
+???
)=(???
×???
)+(???
×???
) 
(3) 
???×???=???×???=???
×???
=0
???×???=???
,???×???
=???,???
×???=?? ·
 
??~
×???
 is easily expressed as a determinant 
???
×???
=|
??? ??? ???
?? 1
?? 2
?? 3
?? 1
?? 2
?? 3
| 
Page 5


 
 
 
VECTOR ANALYSIS
 
 
? Vector analysis is a handy way to write and show math relationships between 
things we measure in the physical world. There are two types of things we look 
at: scalars and vectors.  
 
? Scalars are fully described when we know their size or amount. Examples 
include mass or temperature. 
 
? Vectors need both size and direction to be fully described. Think of things like 
displacement, velocity, or force. 
 
? We write vectors with bold letters or a line above the symbol. When drawn, a 
vector is shown as an arrow. The length of the arrow tells us its size, the direction 
it points tells us its direction, and the arrowhead shows its sense or which way it's 
going. 
Vector Algebra 
? If two-line segments are parallel, have the same length, and go in the same 
direction, we call them equal vectors. If they have the same length but go in 
opposite directions, we call one of them the negative of the other. For example, if 
one vector is shown as "v ? ", its opposite would be "-v ?". 
? If we want a vector that's parallel to " ???” but longer by a certain factor "m", we 
write it as "m·??? ". This means it's "m" times as long as "v ? " and has the same 
direction. 
? Let ???
 and ???
 be any two vectors. 
? if we place the initial point of ???
 on the terminal point of ???
  , the vector ???
, drawn 
from the initial point of ???
 to the terminal point of ???
 is defined as the vector sum of 
???
 and ???
. 
 
???
=???
+???
 
This is the parallelogram law for the composition of two forces (which are vectors). 
Evidently the same vector ?? will be obtained, if we place the initial point of ???
 on the 
terminal point of ????
 : 
 So ???
=???
+???
=???
+???
.  
So, vector addition is commutative (i.e.) the order of vectors appearing in a sum is 
immaterial. The definition of vector addition can be extended to any number of vectors. 
 
 
The addition is also associative, as can be seen from the above diagrams 
???
=(???
+???
)+???
=???
+(???
+???
) 
A unit vector is a vector that has a length of 1.  
Imagine a right-handed rectangular coordinate system.  
Any vector in three dimensions can be shown with its starting point at the origin, which 
we'll call "O".  
Let's say "?? 1
,?? 2
,?? 3
" are the coordinates of the end point of the vector "A" when it starts 
from "O". Let ???,???
 the unit vector in the ?? ,?? ,?? directions respectively, 
 
The vectors ?? 1
,?? 2
?? 1
?? 3
?? , are the rectangular component vectors of ???
 in the ?? ,?? and 
?? ???? rections respectively. The sum or resultant of these is the vector ???
 so that we can 
write 
???
 =?? 1
???+?? 2
??¨+?? 3
???
 
The magnitude of 
???
=v?? 1
2
+?? 2
2
+?? 3
2
=|???
| 
A null vector is a vector (denoted by ???
 ) is a vector whose magnitude is zero. 
Vector Multiplication 
There are two types of multiplication called the dot product and the cross product. 
Dot product 
? The dot product or scalar product of two vectors ???
 and ???
 denoted by ???
·???
 (A dot 
B) defined as the product of the magnitudes of ???
  and ???
 and the cosine of the 
angle between them  
???
·???
=???? cos?? , 0<?? =?? 
???
·???
 Is a scalar 
 
The following results readily follow from the above definition  
(1) ???
·???
=???
·???
 
(2) ???
·(???
+???
)=???
·???
+???
·???
 
(3) ???·???=???·???=???
·???
=1; 
???·???=???·????
=???
·???=0 
(4) If ???
=?? 1
???+???
2
??¨+?? 3
???
 and 
????
=?? 1
???+?? 2
???+?? 3
????
, then  
???
·???
=?? 1
?? 1
+?? 2
?? 2
+?? 3
?? 3
.
???
·???
=?? 1
2
+?? 2
2
+?? 3
 
2
=|???
|
2
???
·???
=?? 1
 
2
+?? 2
2
+?? 3
 
2
=|???
|
2
 
(5) If ???
·????
=0 and ???
 and ????
 are not null vectors, then A and B are perpendicular. 
 
Vector Product 
? The cross or vector product of ???
 and ???
 is a vector ???
=???
×???
 (A cross ?? ).  
? The magnitude of ???
×??~
 is defined as the product of the magnitudes of ???
 and ???
 
and the sine of the angle ?? between them.  
? The direction of the vector ???
=???
×????
 is perpendicular to the plane of ???
 and ???
 and 
such that ???
,???
 and ???
 from a right-handed system. 
???
×???
=???? (sin ?? )???: 0=?? =?? 
? Where ???
 is a unit vector in the direction of ???
×???
. If ???
 is parallel to ???
,??~
×???
=0 
 
The following results are easily deduced: 
(1) ???
×???
=????
¯
×???
 
(2) ???
×(???
+???
)=(???
×???
)+(???
×???
) 
(3) 
???×???=???×???=???
×???
=0
???×???=???
,???×???
=???,???
×???=?? ·
 
??~
×???
 is easily expressed as a determinant 
???
×???
=|
??? ??? ???
?? 1
?? 2
?? 3
?? 1
?? 2
?? 3
| 
 
The magnitude of ???
×???
 is the same as the area of a parallelogram with sides ???
 and ???
. 
Some theorems of geometry and trigonometry are readily derived by vector algebra. 
Some examples are given below: 
 
Example 
Prove the law of cosines for plane triangles. From the figure, 
B
¯
+C
¯
=A
¯
 or C
¯
=A
¯
-B
¯
 ????
·C
¯
=(???
-???
)·(???
-???
)
 =???
·???
+???
·???
-2???
·???
. 
 
i.e., |???
|
2
=|???
|
2
+|???
|
2
-2|???
|·|?? ?
|cos ?? 
 
 
Example 
Derive the trigonometric formula 
cos (?? -?? )=cos ?? cos ?? +sin ?? sin ?? 
Let ?? ,?? be two unit ???? ctors in the ???? plane and let ?? ,?? be the angles they make with the 
?? -axis 
??? =cos ?? ???+sin ?? ???
???
=cos ?? ???+sin ?? ???
 
???·???
=|???||???
|cos (?? -???
)·???? Definition 
Also ???·???
=(cos??????+sir ?? ???)·(cos ?? ???+sin ?? ???)