Scalar and Vector Herts | Mathematics Optional Notes for UPSC PDF Download

 Page 1


Edurev123 
Vector Analysis 
1. Scalar and Vector Herts 
1.1 Prove that the vectors ???? =?? ?? +?? -?? ?? ,???? 
=-?? +?? ?? +?? ?? ,??? =?? ?? -?? ?? -?? ?? can 
form the sides of a triangle. Find the lengths of the medians of the triangle. 
(2016 : 10 Marks) 
Solution: 
Here, we find that 
??? 
+??  =(-?? +3?? +4?? )+(4?? -2?? -6?? )
 =3?? +?? -2?? =?? 
??.?? .,                                                ??? 
+??  =?? 
 
                                                    
And also we notice that these three vectors are not collinear (components are not 
proportional). Hence, these form the sides ofa triangle. Let ???? ,???? and ???? be medians. 
By triangle law of vector addition : 
Page 2


Edurev123 
Vector Analysis 
1. Scalar and Vector Herts 
1.1 Prove that the vectors ???? =?? ?? +?? -?? ?? ,???? 
=-?? +?? ?? +?? ?? ,??? =?? ?? -?? ?? -?? ?? can 
form the sides of a triangle. Find the lengths of the medians of the triangle. 
(2016 : 10 Marks) 
Solution: 
Here, we find that 
??? 
+??  =(-?? +3?? +4?? )+(4?? -2?? -6?? )
 =3?? +?? -2?? =?? 
??.?? .,                                                ??? 
+??  =?? 
 
                                                    
And also we notice that these three vectors are not collinear (components are not 
proportional). Hence, these form the sides ofa triangle. Let ???? ,???? and ???? be medians. 
By triangle law of vector addition : 
????
????? 
=????
????? 
+????
????? 
=?? 
-
?? 
2
=
1
2
(5?? -5?? -14?? )
|?? ?? |
??????? 
=
v
1
4
(25+25+196)=
v
246
4
=
v246
2
????
????? 
=????
????? 
+????
????? 
=-(?? 
+
??? 
2
)=-
1
2
(7??ˆ-??ˆ-8??ˆ
)
 |???? |
??????? 
=
1
2
v49+1+64=
v114
2
????
????? 
=????
????? 
+????
????? 
=??? 
+
?? 
2
=
1
2
(2?? +4?? +2?? )
 |????
????? 
|=|?? +2?? +?? |=v1+4+1=v6
 
  
Page 3


Edurev123 
Vector Analysis 
1. Scalar and Vector Herts 
1.1 Prove that the vectors ???? =?? ?? +?? -?? ?? ,???? 
=-?? +?? ?? +?? ?? ,??? =?? ?? -?? ?? -?? ?? can 
form the sides of a triangle. Find the lengths of the medians of the triangle. 
(2016 : 10 Marks) 
Solution: 
Here, we find that 
??? 
+??  =(-?? +3?? +4?? )+(4?? -2?? -6?? )
 =3?? +?? -2?? =?? 
??.?? .,                                                ??? 
+??  =?? 
 
                                                    
And also we notice that these three vectors are not collinear (components are not 
proportional). Hence, these form the sides ofa triangle. Let ???? ,???? and ???? be medians. 
By triangle law of vector addition : 
????
????? 
=????
????? 
+????
????? 
=?? 
-
?? 
2
=
1
2
(5?? -5?? -14?? )
|?? ?? |
??????? 
=
v
1
4
(25+25+196)=
v
246
4
=
v246
2
????
????? 
=????
????? 
+????
????? 
=-(?? 
+
??? 
2
)=-
1
2
(7??ˆ-??ˆ-8??ˆ
)
 |???? |
??????? 
=
1
2
v49+1+64=
v114
2
????
????? 
=????
????? 
+????
????? 
=??? 
+
?? 
2
=
1
2
(2?? +4?? +2?? )
 |????
????? 
|=|?? +2?? +?? |=v1+4+1=v6
 
  
Edurev123 
2. Differentiation of a Vector Field of a 
Scalar Variable 
2.1 For two vectors ????  and ???? 
 given respectively by 
and 
???? =?? ?? ?? ??ˆ+?? ??ˆ-?? ?? ??ˆ
???? 
=?????? ?? ??ˆ-?????? ?? ??ˆ
 
Determine: (i) 
?? ????
(???? ·???? 
) and 
?? ????
(???? ×???? 
) 
(2009: 10 marks) 
Solution: 
??  =5?? 2
??ˆ+?? ??ˆ-?? 3
??ˆ
??? 
 =sin 5?? ??ˆ-cos ?? ??ˆ
?? ·??? 
 =5?? 2
sin 5?? -?? cos ?? (???ˆ·??ˆ=1 etc., ??ˆ·??ˆ=??ˆ·??ˆ
=??ˆ
·??ˆ=0)
?                        
?? ????
(?? ·??? 
) =
?? ????
(5?? 2
sin 5?? -?? cos ?? )
 =5(2?? sin 5?? +?? 2
·5cos 5?? )-(1·cos ?? -?? sin ?? )
 =10?? sin 5?? +25?? 2
cos 5?? -cos ?? +?? sin ?? 
?? ×??? 
 =|
??ˆ ??ˆ ??ˆ
5?? 2
?? -?? 3
sin 5?? -cos ?? 0
|
=??ˆ(0-?? 3
cos ?? )+??ˆ(-?? 3
sin 5?? -0)+??ˆ
(-5?? 2
cos ?? -??ˆ
sin 5?? )
= -?? 3
cos ?? ??ˆ-?? 3
sin 5?? ??ˆ-(5?? 2
cos ?? +?? sin 5?? )??ˆ
?                    
?? ????
(?? ×??? 
)=(-3?? 2
cos ?? +?? 3
sin ?? )??ˆ-(3?? 2
sin 5?? +5?? 3
cos ?? )??ˆ-
(10?? cos ?? -5?? 2
sin ?? +?? cos 5?? +1·sin 5?? )??ˆ
 
2.2 If 
???? 
=?? ?? ???? ?? -?? ?? ?? ?? ?? +?? ?? ?? ???? 
???? 
=?? ?? ?? +?? ?? -?? ?? ???? 
 
find the value of 
?? ?? ?? ?? ?? ?? (???? 
×???? 
) at (?? ,?? ,-?? ) . 
(2012 : 12 Marks) 
Solution: 
Page 4


Edurev123 
Vector Analysis 
1. Scalar and Vector Herts 
1.1 Prove that the vectors ???? =?? ?? +?? -?? ?? ,???? 
=-?? +?? ?? +?? ?? ,??? =?? ?? -?? ?? -?? ?? can 
form the sides of a triangle. Find the lengths of the medians of the triangle. 
(2016 : 10 Marks) 
Solution: 
Here, we find that 
??? 
+??  =(-?? +3?? +4?? )+(4?? -2?? -6?? )
 =3?? +?? -2?? =?? 
??.?? .,                                                ??? 
+??  =?? 
 
                                                    
And also we notice that these three vectors are not collinear (components are not 
proportional). Hence, these form the sides ofa triangle. Let ???? ,???? and ???? be medians. 
By triangle law of vector addition : 
????
????? 
=????
????? 
+????
????? 
=?? 
-
?? 
2
=
1
2
(5?? -5?? -14?? )
|?? ?? |
??????? 
=
v
1
4
(25+25+196)=
v
246
4
=
v246
2
????
????? 
=????
????? 
+????
????? 
=-(?? 
+
??? 
2
)=-
1
2
(7??ˆ-??ˆ-8??ˆ
)
 |???? |
??????? 
=
1
2
v49+1+64=
v114
2
????
????? 
=????
????? 
+????
????? 
=??? 
+
?? 
2
=
1
2
(2?? +4?? +2?? )
 |????
????? 
|=|?? +2?? +?? |=v1+4+1=v6
 
  
Edurev123 
2. Differentiation of a Vector Field of a 
Scalar Variable 
2.1 For two vectors ????  and ???? 
 given respectively by 
and 
???? =?? ?? ?? ??ˆ+?? ??ˆ-?? ?? ??ˆ
???? 
=?????? ?? ??ˆ-?????? ?? ??ˆ
 
Determine: (i) 
?? ????
(???? ·???? 
) and 
?? ????
(???? ×???? 
) 
(2009: 10 marks) 
Solution: 
??  =5?? 2
??ˆ+?? ??ˆ-?? 3
??ˆ
??? 
 =sin 5?? ??ˆ-cos ?? ??ˆ
?? ·??? 
 =5?? 2
sin 5?? -?? cos ?? (???ˆ·??ˆ=1 etc., ??ˆ·??ˆ=??ˆ·??ˆ
=??ˆ
·??ˆ=0)
?                        
?? ????
(?? ·??? 
) =
?? ????
(5?? 2
sin 5?? -?? cos ?? )
 =5(2?? sin 5?? +?? 2
·5cos 5?? )-(1·cos ?? -?? sin ?? )
 =10?? sin 5?? +25?? 2
cos 5?? -cos ?? +?? sin ?? 
?? ×??? 
 =|
??ˆ ??ˆ ??ˆ
5?? 2
?? -?? 3
sin 5?? -cos ?? 0
|
=??ˆ(0-?? 3
cos ?? )+??ˆ(-?? 3
sin 5?? -0)+??ˆ
(-5?? 2
cos ?? -??ˆ
sin 5?? )
= -?? 3
cos ?? ??ˆ-?? 3
sin 5?? ??ˆ-(5?? 2
cos ?? +?? sin 5?? )??ˆ
?                    
?? ????
(?? ×??? 
)=(-3?? 2
cos ?? +?? 3
sin ?? )??ˆ-(3?? 2
sin 5?? +5?? 3
cos ?? )??ˆ-
(10?? cos ?? -5?? 2
sin ?? +?? cos 5?? +1·sin 5?? )??ˆ
 
2.2 If 
???? 
=?? ?? ???? ?? -?? ?? ?? ?? ?? +?? ?? ?? ???? 
???? 
=?? ?? ?? +?? ?? -?? ?? ???? 
 
find the value of 
?? ?? ?? ?? ?? ?? (???? 
×???? 
) at (?? ,?? ,-?? ) . 
(2012 : 12 Marks) 
Solution: 
Given: 
                    ?? ???????????????????????? 
=?? 2
???? ?? -2?? ?? 3
?? +?? ?? 2
??? 
                   ?? ??????????????????????? 
=2?? ?? +?? ?? -?? 2
??? 
 ?      ?? 
×??? 
=|
?? ?? ??? 
?? 2
???? -2?? ?? 3
?? ?? 2
2?? ?? -?? 2
|
                       =?? (2?? 3
?? 3
-???? ?? 2
)+?? (2?? ?? 3
+?? 4
???? )+??? 
(?? 2
?? 2
?? +4?? ?? 4
)
 
?
??? (?? 
×??? 
)=?? (-?? ?? 2
)+?? (?? 4
?? )+??? 
(2?? 2
???? )
 ?           
?
2
??? ??? (?? 
×??? 
)=?? (-?? 2
)+?? (4?? 3
?? )+??? 
(4?????? )
 ? At (1,0,-2) ? 
                   
?
2
??? ??? (?? 
×??? 
)=-4?? -8?? 
 
2.3 The position vector of a moving point at time ?? is, ?? 
=(?????? ?? )?? +(?????? ?? ?? )?? +
(?? ?? +?? ?? )?? . Find the components of acceleration ????  in the directions parallel to the 
velocity vector ????  and perpendicular to the plane of ???  and ????  at time ?? =?? . 
(2017 : 10 Marks) 
Solution: 
?? =(sin ?? )?? +(cos 2?? )?? +(?? 2
+2?? )?? (??)
?? =
?? ?? 
????
=(cos ?? )?? -2sin 2???? +(2?? +2)?? (???? )
?? =
?? 2
?? 
?? ?? 2
=(-sin ?? )?? -4cos 2???? +2?? (?????? )
 
At ?? =0, 
?? =?? ,?? =?? +2?? ,?? =-4?? ÷2?? 
Component of ??  in direction of ??  
?? 
?? =
?? ·?? 
|?? |
·
?? 
|?? |
=
+4
(1+4)
?? =
4
5
(?? +2?? ) 
Component of ??  in the direction of vector perpendicular to the plane of ??  and ?? . 
Let ??  be the vector perpendicular to ??  and ??  
??  =?? ×?? =2??ˆ-??ˆ
,|?? |
2
=5
?? 
?? =
?? ·?? 
|??? 
|
·
??? 
|?? |
=
-2
5
(2?? -?? )