Detailed Notes: Vector Algebra | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


Vectors & Three Dimensional 
Geometry 
Vectors and their Representation: 
 
Vector quantities are specified by definite magnitude and definite direction. A vector is 
generally represented by a directed line segment, say ????
????? 
. ?? is called the initial point and 
?? is called the terminal point. The magnitude of vector ????
????? 
 is expressed by |????
????? 
|. 
Zero vector: 
A vector of zero magnitude i.e. which has the same initial and terminal point, is called a 
zero vector. It is denoted by ?? . The direction of zero vector is indeterminate. 
Unit vector: 
A vector of unit magnitude in the direction of a vector ??  is called unit vector along ??  and 
is denoted by ?? , symbolically ?? ˆ =
??? 
|??? |
. 
Equal vectors: 
Two vectors are said to be equal if they have the same magnitude, direction and 
represent the same physical quantity. 
Collinear vectors: 
Two vectors are said to be collinear if their directed line segments are parallel 
irrespective of their directions. Collinear vectors are also called parallel vectors. If they 
have the same direction (?) 
they are named as like vectors but if they have opposite direction (?) then they are 
named as unlike vectors. 
Symbolically, two non-zero vectors ??  and ?? ? 
 are collinear if and only if, ?? = ?? ?? ? 
, where ?? ?
?? 
Page 2


Vectors & Three Dimensional 
Geometry 
Vectors and their Representation: 
 
Vector quantities are specified by definite magnitude and definite direction. A vector is 
generally represented by a directed line segment, say ????
????? 
. ?? is called the initial point and 
?? is called the terminal point. The magnitude of vector ????
????? 
 is expressed by |????
????? 
|. 
Zero vector: 
A vector of zero magnitude i.e. which has the same initial and terminal point, is called a 
zero vector. It is denoted by ?? . The direction of zero vector is indeterminate. 
Unit vector: 
A vector of unit magnitude in the direction of a vector ??  is called unit vector along ??  and 
is denoted by ?? , symbolically ?? ˆ =
??? 
|??? |
. 
Equal vectors: 
Two vectors are said to be equal if they have the same magnitude, direction and 
represent the same physical quantity. 
Collinear vectors: 
Two vectors are said to be collinear if their directed line segments are parallel 
irrespective of their directions. Collinear vectors are also called parallel vectors. If they 
have the same direction (?) 
they are named as like vectors but if they have opposite direction (?) then they are 
named as unlike vectors. 
Symbolically, two non-zero vectors ??  and ?? ? 
 are collinear if and only if, ?? = ?? ?? ? 
, where ?? ?
?? 
?? = ?? ?? ? 
? (?? 1
?? ˆ + ?? 2
?? ˆ + ?? 3
?? ˆ)= ?? (?? 1
?? ˆ + ?? 2
?? ˆ + ?? 3
?? ˆ)? ?? 1
= ?? ?? 1
,?? 2
= ?? ?? 2
,?? 3
= ?? ?? 3
?
?? 1
?? 1
=
?? 2
?? 2
=
?? 3
?? 3
 (= ?? ) 
Vectors ?? = ?? 1
?? ˆ + ?? 2
?? ˆ + ?? 3
?? ˆ and ?? ? 
= ?? 1
?? ˆ + ?? 2
?? ˆ + ?? 3
?? ˆ are collinear if 
?? 1
?? 1
=
?? 2
?? 2
=
?? 3
?? 3
 
Note: If ?? ,?? ? 
 are non zero, non-collinear vectors, such that ?? ?? + ?? ?? ? 
= ?? '
?? + ?? '
?? ? 
? ?? =
?? '
,?? = ?? '
, (where ?? ,?? '
,?? ,?? ' are scalars) 
Problem 1: Find unit vector of ?? ˆ - 2?? ˆ + 3?? ˆ 
Solution:  ?? = ?? ˆ - 2?? ˆ + 3?? ˆ 
 ????  ?? = ?? ?? ?? ˆ + ?? ?? ?? ˆ + ?? ?? ?? ˆ  ?? h???? |?? | = v?? ?? 2
+ ?? ?? 2
+ ?? ?? 2
 ? |?? | = v14 ? ?? ˆ =
?? 
|?? |
=
1
v14
?? ˆ -
2
v14
?? ˆ +
3
v14
?? ˆ  
Problem 2: ?? = (?? + 1)?? ˆ - (2?? + ?? )?? ˆ + 3?? ˆ and ?? ? 
= (2?? - 1)?? ˆ + (2+ 3?? )?? ˆ + ?? ˆ find ??  
and ??  for which ??  and ?? ? 
 are parallel. 
Solution:  ??  and ?? ? 
 are parallel ? 
?? +1
2?? -1
=
-(2?? +?? )
2+3?? =
3
1
? ?? = 4/5,?? = -19/25 
Coplanar vectors: 
A given number vectors are called coplanar if their line segments are all parallel to the 
same plane. Note that "two vectors are always coplanar". 
Multiplication of a vector by a scalar: 
If ??  is a vector and ?? is a scalar, then ?? is a vector parallel to a whose magnitude is |?? | 
times that of ?? . This multiplication is called scalar multiplication. If ??  and ??  are vectors 
and ?? ,?? are scalars, then: 
(i) 
?? (?? )= (?? )?? = ?? ??  
(ii) 
?? (?? ?? )= ?? (?? ?? )= (???? )??  
(iii)  (?? + ?? )?? = ?? ?? + ?? ??  
(iv)  ?? (?? + ?? ? 
)= ?? ?? + ?? ?? ? 
 
Addition of vectors: 
(i) If two vectors ??  and ?? ? 
 are represented by ????
????? 
 and ????
????? 
, then their sum ?? + ?? ? 
 is a vector 
represented by ????
????? 
, where ???? is the diagonal of the parallelogram OACB. 
(ii) ?? + ?? ? 
= ?? ? 
+ ??  (commutative) (iii)  (?? + ?? ? 
)+ ?? = ?? + (?? ? 
+ ?? )  (associative) 
Page 3


Vectors & Three Dimensional 
Geometry 
Vectors and their Representation: 
 
Vector quantities are specified by definite magnitude and definite direction. A vector is 
generally represented by a directed line segment, say ????
????? 
. ?? is called the initial point and 
?? is called the terminal point. The magnitude of vector ????
????? 
 is expressed by |????
????? 
|. 
Zero vector: 
A vector of zero magnitude i.e. which has the same initial and terminal point, is called a 
zero vector. It is denoted by ?? . The direction of zero vector is indeterminate. 
Unit vector: 
A vector of unit magnitude in the direction of a vector ??  is called unit vector along ??  and 
is denoted by ?? , symbolically ?? ˆ =
??? 
|??? |
. 
Equal vectors: 
Two vectors are said to be equal if they have the same magnitude, direction and 
represent the same physical quantity. 
Collinear vectors: 
Two vectors are said to be collinear if their directed line segments are parallel 
irrespective of their directions. Collinear vectors are also called parallel vectors. If they 
have the same direction (?) 
they are named as like vectors but if they have opposite direction (?) then they are 
named as unlike vectors. 
Symbolically, two non-zero vectors ??  and ?? ? 
 are collinear if and only if, ?? = ?? ?? ? 
, where ?? ?
?? 
?? = ?? ?? ? 
? (?? 1
?? ˆ + ?? 2
?? ˆ + ?? 3
?? ˆ)= ?? (?? 1
?? ˆ + ?? 2
?? ˆ + ?? 3
?? ˆ)? ?? 1
= ?? ?? 1
,?? 2
= ?? ?? 2
,?? 3
= ?? ?? 3
?
?? 1
?? 1
=
?? 2
?? 2
=
?? 3
?? 3
 (= ?? ) 
Vectors ?? = ?? 1
?? ˆ + ?? 2
?? ˆ + ?? 3
?? ˆ and ?? ? 
= ?? 1
?? ˆ + ?? 2
?? ˆ + ?? 3
?? ˆ are collinear if 
?? 1
?? 1
=
?? 2
?? 2
=
?? 3
?? 3
 
Note: If ?? ,?? ? 
 are non zero, non-collinear vectors, such that ?? ?? + ?? ?? ? 
= ?? '
?? + ?? '
?? ? 
? ?? =
?? '
,?? = ?? '
, (where ?? ,?? '
,?? ,?? ' are scalars) 
Problem 1: Find unit vector of ?? ˆ - 2?? ˆ + 3?? ˆ 
Solution:  ?? = ?? ˆ - 2?? ˆ + 3?? ˆ 
 ????  ?? = ?? ?? ?? ˆ + ?? ?? ?? ˆ + ?? ?? ?? ˆ  ?? h???? |?? | = v?? ?? 2
+ ?? ?? 2
+ ?? ?? 2
 ? |?? | = v14 ? ?? ˆ =
?? 
|?? |
=
1
v14
?? ˆ -
2
v14
?? ˆ +
3
v14
?? ˆ  
Problem 2: ?? = (?? + 1)?? ˆ - (2?? + ?? )?? ˆ + 3?? ˆ and ?? ? 
= (2?? - 1)?? ˆ + (2+ 3?? )?? ˆ + ?? ˆ find ??  
and ??  for which ??  and ?? ? 
 are parallel. 
Solution:  ??  and ?? ? 
 are parallel ? 
?? +1
2?? -1
=
-(2?? +?? )
2+3?? =
3
1
? ?? = 4/5,?? = -19/25 
Coplanar vectors: 
A given number vectors are called coplanar if their line segments are all parallel to the 
same plane. Note that "two vectors are always coplanar". 
Multiplication of a vector by a scalar: 
If ??  is a vector and ?? is a scalar, then ?? is a vector parallel to a whose magnitude is |?? | 
times that of ?? . This multiplication is called scalar multiplication. If ??  and ??  are vectors 
and ?? ,?? are scalars, then: 
(i) 
?? (?? )= (?? )?? = ?? ??  
(ii) 
?? (?? ?? )= ?? (?? ?? )= (???? )??  
(iii)  (?? + ?? )?? = ?? ?? + ?? ??  
(iv)  ?? (?? + ?? ? 
)= ?? ?? + ?? ?? ? 
 
Addition of vectors: 
(i) If two vectors ??  and ?? ? 
 are represented by ????
????? 
 and ????
????? 
, then their sum ?? + ?? ? 
 is a vector 
represented by ????
????? 
, where ???? is the diagonal of the parallelogram OACB. 
(ii) ?? + ?? ? 
= ?? ? 
+ ??  (commutative) (iii)  (?? + ?? ? 
)+ ?? = ?? + (?? ? 
+ ?? )  (associative) 
(iv) ?? + 0
? 
= ?? = 0
? 
+ ??  
(v)  ?? + (-?? )= 0
? 
= (-?? )+ ??  
(vi)  |?? + ?? ? 
| = |?? | + |?? ? 
| 
(vii)  |?? - ?? ? 
| = ||?? | - |?? ? 
|| 
Problem 3: The two sides of ? ?????? are given by ????
????? 
= 2?? ˆ + 4?? ˆ + 4?? ˆ,????
????? 
= 2?? ˆ + 2?? ˆ + ?? ˆ. 
Then find the length of median through A. 
Solution: Let ?? be mid point of ???? 
  ???? ? ?????? ,????
????? 
+ ????
?????? 
= ????
????? 
 ? ????
????? 
+
1
2
????
????? 
= ????
????? 
   ? 
????
????? 
+ (????
????? 
+ ????
????? 
)
2
= ????
????? 
   ? 
????
????? 
+ ????
????? 
2
= ????
????? 
? |????
????? 
| = |
4?? ˆ + 6?? ˆ + 5?? ˆ
2
| =
v77
2
  
 
Problem 4: In a triangle ?????? ,?? ,?? ,?? are the mid-points of the sides ???? ,???? and ???? 
respectively then prove that, ????
????? 
= -(????
????? 
+ ????
????? 
) . 
Solution: 
????
????? 
= 3????
????? 
  = 3 ·
1
2
(????
????? 
+ ????
????? 
) ?? h?????? ?? ???? ?????? - ?????????? ???? ????   =
3
2
[
2
3
????
????? 
+
2
3
????
????? 
]
= -(????
????? 
+ ????
????? 
)  
Page 4


Vectors & Three Dimensional 
Geometry 
Vectors and their Representation: 
 
Vector quantities are specified by definite magnitude and definite direction. A vector is 
generally represented by a directed line segment, say ????
????? 
. ?? is called the initial point and 
?? is called the terminal point. The magnitude of vector ????
????? 
 is expressed by |????
????? 
|. 
Zero vector: 
A vector of zero magnitude i.e. which has the same initial and terminal point, is called a 
zero vector. It is denoted by ?? . The direction of zero vector is indeterminate. 
Unit vector: 
A vector of unit magnitude in the direction of a vector ??  is called unit vector along ??  and 
is denoted by ?? , symbolically ?? ˆ =
??? 
|??? |
. 
Equal vectors: 
Two vectors are said to be equal if they have the same magnitude, direction and 
represent the same physical quantity. 
Collinear vectors: 
Two vectors are said to be collinear if their directed line segments are parallel 
irrespective of their directions. Collinear vectors are also called parallel vectors. If they 
have the same direction (?) 
they are named as like vectors but if they have opposite direction (?) then they are 
named as unlike vectors. 
Symbolically, two non-zero vectors ??  and ?? ? 
 are collinear if and only if, ?? = ?? ?? ? 
, where ?? ?
?? 
?? = ?? ?? ? 
? (?? 1
?? ˆ + ?? 2
?? ˆ + ?? 3
?? ˆ)= ?? (?? 1
?? ˆ + ?? 2
?? ˆ + ?? 3
?? ˆ)? ?? 1
= ?? ?? 1
,?? 2
= ?? ?? 2
,?? 3
= ?? ?? 3
?
?? 1
?? 1
=
?? 2
?? 2
=
?? 3
?? 3
 (= ?? ) 
Vectors ?? = ?? 1
?? ˆ + ?? 2
?? ˆ + ?? 3
?? ˆ and ?? ? 
= ?? 1
?? ˆ + ?? 2
?? ˆ + ?? 3
?? ˆ are collinear if 
?? 1
?? 1
=
?? 2
?? 2
=
?? 3
?? 3
 
Note: If ?? ,?? ? 
 are non zero, non-collinear vectors, such that ?? ?? + ?? ?? ? 
= ?? '
?? + ?? '
?? ? 
? ?? =
?? '
,?? = ?? '
, (where ?? ,?? '
,?? ,?? ' are scalars) 
Problem 1: Find unit vector of ?? ˆ - 2?? ˆ + 3?? ˆ 
Solution:  ?? = ?? ˆ - 2?? ˆ + 3?? ˆ 
 ????  ?? = ?? ?? ?? ˆ + ?? ?? ?? ˆ + ?? ?? ?? ˆ  ?? h???? |?? | = v?? ?? 2
+ ?? ?? 2
+ ?? ?? 2
 ? |?? | = v14 ? ?? ˆ =
?? 
|?? |
=
1
v14
?? ˆ -
2
v14
?? ˆ +
3
v14
?? ˆ  
Problem 2: ?? = (?? + 1)?? ˆ - (2?? + ?? )?? ˆ + 3?? ˆ and ?? ? 
= (2?? - 1)?? ˆ + (2+ 3?? )?? ˆ + ?? ˆ find ??  
and ??  for which ??  and ?? ? 
 are parallel. 
Solution:  ??  and ?? ? 
 are parallel ? 
?? +1
2?? -1
=
-(2?? +?? )
2+3?? =
3
1
? ?? = 4/5,?? = -19/25 
Coplanar vectors: 
A given number vectors are called coplanar if their line segments are all parallel to the 
same plane. Note that "two vectors are always coplanar". 
Multiplication of a vector by a scalar: 
If ??  is a vector and ?? is a scalar, then ?? is a vector parallel to a whose magnitude is |?? | 
times that of ?? . This multiplication is called scalar multiplication. If ??  and ??  are vectors 
and ?? ,?? are scalars, then: 
(i) 
?? (?? )= (?? )?? = ?? ??  
(ii) 
?? (?? ?? )= ?? (?? ?? )= (???? )??  
(iii)  (?? + ?? )?? = ?? ?? + ?? ??  
(iv)  ?? (?? + ?? ? 
)= ?? ?? + ?? ?? ? 
 
Addition of vectors: 
(i) If two vectors ??  and ?? ? 
 are represented by ????
????? 
 and ????
????? 
, then their sum ?? + ?? ? 
 is a vector 
represented by ????
????? 
, where ???? is the diagonal of the parallelogram OACB. 
(ii) ?? + ?? ? 
= ?? ? 
+ ??  (commutative) (iii)  (?? + ?? ? 
)+ ?? = ?? + (?? ? 
+ ?? )  (associative) 
(iv) ?? + 0
? 
= ?? = 0
? 
+ ??  
(v)  ?? + (-?? )= 0
? 
= (-?? )+ ??  
(vi)  |?? + ?? ? 
| = |?? | + |?? ? 
| 
(vii)  |?? - ?? ? 
| = ||?? | - |?? ? 
|| 
Problem 3: The two sides of ? ?????? are given by ????
????? 
= 2?? ˆ + 4?? ˆ + 4?? ˆ,????
????? 
= 2?? ˆ + 2?? ˆ + ?? ˆ. 
Then find the length of median through A. 
Solution: Let ?? be mid point of ???? 
  ???? ? ?????? ,????
????? 
+ ????
?????? 
= ????
????? 
 ? ????
????? 
+
1
2
????
????? 
= ????
????? 
   ? 
????
????? 
+ (????
????? 
+ ????
????? 
)
2
= ????
????? 
   ? 
????
????? 
+ ????
????? 
2
= ????
????? 
? |????
????? 
| = |
4?? ˆ + 6?? ˆ + 5?? ˆ
2
| =
v77
2
  
 
Problem 4: In a triangle ?????? ,?? ,?? ,?? are the mid-points of the sides ???? ,???? and ???? 
respectively then prove that, ????
????? 
= -(????
????? 
+ ????
????? 
) . 
Solution: 
????
????? 
= 3????
????? 
  = 3 ·
1
2
(????
????? 
+ ????
????? 
) ?? h?????? ?? ???? ?????? - ?????????? ???? ????   =
3
2
[
2
3
????
????? 
+
2
3
????
????? 
]
= -(????
????? 
+ ????
????? 
)  
 
Position vector of a point: 
Let ?? be a fixed origin, then the position vector of a point ?? is the vector ????
????? 
. If ??  and ?? ? 
 
are position vectors of two points ?? and ?? , then 
????
????? 
= ?? ? 
- ?? = ???????????????? ???????????? (?? .?? .) ???? ?? - ???????????????? ???????????? (?? .?? .) ???? ?? .  
DISTANCE FORMULA 
Distance between the two points ?? (?? ) and ?? (?? ? 
) is ???? = |?? - ?? ? 
| 
SECTION FORMULA 
 
If ??  and ?? ? 
 are the position vectors of two points ?? (?? 1
,?? 1
,?? 1
) and ?? (?? 2
,?? 2
,?? 2
) , then the 
p.v. of a point ?? which divides ???? in the ratio ?? :?? is given by ?? =
?? ??? +?? ?? ? 
?? +?? 
Page 5


Vectors & Three Dimensional 
Geometry 
Vectors and their Representation: 
 
Vector quantities are specified by definite magnitude and definite direction. A vector is 
generally represented by a directed line segment, say ????
????? 
. ?? is called the initial point and 
?? is called the terminal point. The magnitude of vector ????
????? 
 is expressed by |????
????? 
|. 
Zero vector: 
A vector of zero magnitude i.e. which has the same initial and terminal point, is called a 
zero vector. It is denoted by ?? . The direction of zero vector is indeterminate. 
Unit vector: 
A vector of unit magnitude in the direction of a vector ??  is called unit vector along ??  and 
is denoted by ?? , symbolically ?? ˆ =
??? 
|??? |
. 
Equal vectors: 
Two vectors are said to be equal if they have the same magnitude, direction and 
represent the same physical quantity. 
Collinear vectors: 
Two vectors are said to be collinear if their directed line segments are parallel 
irrespective of their directions. Collinear vectors are also called parallel vectors. If they 
have the same direction (?) 
they are named as like vectors but if they have opposite direction (?) then they are 
named as unlike vectors. 
Symbolically, two non-zero vectors ??  and ?? ? 
 are collinear if and only if, ?? = ?? ?? ? 
, where ?? ?
?? 
?? = ?? ?? ? 
? (?? 1
?? ˆ + ?? 2
?? ˆ + ?? 3
?? ˆ)= ?? (?? 1
?? ˆ + ?? 2
?? ˆ + ?? 3
?? ˆ)? ?? 1
= ?? ?? 1
,?? 2
= ?? ?? 2
,?? 3
= ?? ?? 3
?
?? 1
?? 1
=
?? 2
?? 2
=
?? 3
?? 3
 (= ?? ) 
Vectors ?? = ?? 1
?? ˆ + ?? 2
?? ˆ + ?? 3
?? ˆ and ?? ? 
= ?? 1
?? ˆ + ?? 2
?? ˆ + ?? 3
?? ˆ are collinear if 
?? 1
?? 1
=
?? 2
?? 2
=
?? 3
?? 3
 
Note: If ?? ,?? ? 
 are non zero, non-collinear vectors, such that ?? ?? + ?? ?? ? 
= ?? '
?? + ?? '
?? ? 
? ?? =
?? '
,?? = ?? '
, (where ?? ,?? '
,?? ,?? ' are scalars) 
Problem 1: Find unit vector of ?? ˆ - 2?? ˆ + 3?? ˆ 
Solution:  ?? = ?? ˆ - 2?? ˆ + 3?? ˆ 
 ????  ?? = ?? ?? ?? ˆ + ?? ?? ?? ˆ + ?? ?? ?? ˆ  ?? h???? |?? | = v?? ?? 2
+ ?? ?? 2
+ ?? ?? 2
 ? |?? | = v14 ? ?? ˆ =
?? 
|?? |
=
1
v14
?? ˆ -
2
v14
?? ˆ +
3
v14
?? ˆ  
Problem 2: ?? = (?? + 1)?? ˆ - (2?? + ?? )?? ˆ + 3?? ˆ and ?? ? 
= (2?? - 1)?? ˆ + (2+ 3?? )?? ˆ + ?? ˆ find ??  
and ??  for which ??  and ?? ? 
 are parallel. 
Solution:  ??  and ?? ? 
 are parallel ? 
?? +1
2?? -1
=
-(2?? +?? )
2+3?? =
3
1
? ?? = 4/5,?? = -19/25 
Coplanar vectors: 
A given number vectors are called coplanar if their line segments are all parallel to the 
same plane. Note that "two vectors are always coplanar". 
Multiplication of a vector by a scalar: 
If ??  is a vector and ?? is a scalar, then ?? is a vector parallel to a whose magnitude is |?? | 
times that of ?? . This multiplication is called scalar multiplication. If ??  and ??  are vectors 
and ?? ,?? are scalars, then: 
(i) 
?? (?? )= (?? )?? = ?? ??  
(ii) 
?? (?? ?? )= ?? (?? ?? )= (???? )??  
(iii)  (?? + ?? )?? = ?? ?? + ?? ??  
(iv)  ?? (?? + ?? ? 
)= ?? ?? + ?? ?? ? 
 
Addition of vectors: 
(i) If two vectors ??  and ?? ? 
 are represented by ????
????? 
 and ????
????? 
, then their sum ?? + ?? ? 
 is a vector 
represented by ????
????? 
, where ???? is the diagonal of the parallelogram OACB. 
(ii) ?? + ?? ? 
= ?? ? 
+ ??  (commutative) (iii)  (?? + ?? ? 
)+ ?? = ?? + (?? ? 
+ ?? )  (associative) 
(iv) ?? + 0
? 
= ?? = 0
? 
+ ??  
(v)  ?? + (-?? )= 0
? 
= (-?? )+ ??  
(vi)  |?? + ?? ? 
| = |?? | + |?? ? 
| 
(vii)  |?? - ?? ? 
| = ||?? | - |?? ? 
|| 
Problem 3: The two sides of ? ?????? are given by ????
????? 
= 2?? ˆ + 4?? ˆ + 4?? ˆ,????
????? 
= 2?? ˆ + 2?? ˆ + ?? ˆ. 
Then find the length of median through A. 
Solution: Let ?? be mid point of ???? 
  ???? ? ?????? ,????
????? 
+ ????
?????? 
= ????
????? 
 ? ????
????? 
+
1
2
????
????? 
= ????
????? 
   ? 
????
????? 
+ (????
????? 
+ ????
????? 
)
2
= ????
????? 
   ? 
????
????? 
+ ????
????? 
2
= ????
????? 
? |????
????? 
| = |
4?? ˆ + 6?? ˆ + 5?? ˆ
2
| =
v77
2
  
 
Problem 4: In a triangle ?????? ,?? ,?? ,?? are the mid-points of the sides ???? ,???? and ???? 
respectively then prove that, ????
????? 
= -(????
????? 
+ ????
????? 
) . 
Solution: 
????
????? 
= 3????
????? 
  = 3 ·
1
2
(????
????? 
+ ????
????? 
) ?? h?????? ?? ???? ?????? - ?????????? ???? ????   =
3
2
[
2
3
????
????? 
+
2
3
????
????? 
]
= -(????
????? 
+ ????
????? 
)  
 
Position vector of a point: 
Let ?? be a fixed origin, then the position vector of a point ?? is the vector ????
????? 
. If ??  and ?? ? 
 
are position vectors of two points ?? and ?? , then 
????
????? 
= ?? ? 
- ?? = ???????????????? ???????????? (?? .?? .) ???? ?? - ???????????????? ???????????? (?? .?? .) ???? ?? .  
DISTANCE FORMULA 
Distance between the two points ?? (?? ) and ?? (?? ? 
) is ???? = |?? - ?? ? 
| 
SECTION FORMULA 
 
If ??  and ?? ? 
 are the position vectors of two points ?? (?? 1
,?? 1
,?? 1
) and ?? (?? 2
,?? 2
,?? 2
) , then the 
p.v. of a point ?? which divides ???? in the ratio ?? :?? is given by ?? =
?? ??? +?? ?? ? 
?? +?? 
Here ?? = (
?? ?? 1
+?? ?? 2
?? +?? ,
?? ?? 1
+?? ?? 2
?? +?? ,
?? ?? 1
+?? ?? 2
?? +?? ) 
 
Note: Position vector of mid point ?? of ???? is 
??? +?? ? 
2
. Here ?? = (
?? 1
+?? 2
2
,
?? 1
+?? 2
2
,
?? 1
+?? 2
2
) 
Problem 5: Let ?? be the centre of a regular pentagon ?????????? and ????
????? 
= ?? . 
Then ????
????? 
+ 2????
????? 
+ 3????
????? 
+ 4????
????? 
+ 5????
????? 
= 
Solution:  ????
????? 
= ?? ,????
????? 
= ?? ? 
,????
????? 
= ?? ,????
?????? 
= ?? 
, ????
????? 
= ??  
????
????? 
+ 2????
????? 
+ 3????
????? 
+ 4????
????? 
+ 5????
????? 
= (?? ? 
- ?? )+ 2(?? - ?? ? 
)+ 3(?? 
- ?? )+ 4(?? - ?? 
)+ 5(?? - ?? ) 
= 5?? - (?? + ?? ? 
+ ?? + ?? 
+ ?? )= 5?? ,( since ?? + ?? ? 
+ ?? + ?? 
+ +?? = 0) 
Problem 6: In a triangle ?????? ,?? and ?? are points on ???? and ???? respectively, such that 
???? = 2???? and ???? = 3???? . Let ?? be the point of intersection of ???? and ???? . Find 
????
????
 using 
vector method. 
Solution: Let the position vectors of points ?? and ?? be respectively ?? ? 
 and ??  referred to ?? 
as origin of reference. 
Let 
????
????
= ??  and  
????
????
= ?? 
????
????? 
=
2?? + ?? ? 
3
,????
????? 
=
3
4
?? ? ????
????? 
=
3?? ?? 
4
+ ?? ? 
?? + 1
=
2?? + ?? ? 
3
?? + 1
 
comparing the coefficient of ?? ? 
 & ??