10

Chapter
VECTOR ALGEBRA

NCERT CRUX

A unit vector in the direction of a is denoted by â. Thus
Scalar and Vector Quantities 
a vector a
â =
=
a magnitude of a
A physical quantity which is completely specified by its
magnitude only is called scalar. It is represented by a real Note:
number along with suitable unit.
(i) â = 1
For example: Distance, Mass, Length, Time, Volume, Speed,
(ii) Unit vectors parallel to x-axis, y-axis and z-axis
Area are scalars quantities.
are denoted by ˆi, ˆj and k̂ respectively.
On the other hand, a physical quantity which has magnitude as  
well as direction is called a vector. 3. Equal Vector: Two vectors a and b and are said to be
For example: Displacement, velocity, acceleration, force etc. equal, if
 
are vector quantities. (a) a = b
(b) They have the same direction.
4. Collinear Vectors or Parallel Vector: Vectors which
Vector
are parallel to the same line are called collinear vectors
or parallel vectors. Such vectors have either same
A line that has magnitude as well as direction is known as
direction or opposite direction.
directed line segment and a directed line segment is called a
vector. If they have the same direction they are said to be like
  vectors, and if they have opposite directions, they are
It is denoted by a or AB , where A & B are initial and terminal
called unlike vectors.
points respectively.

Position Vector

Position vector of a point P (x,y, z) is a vector whose initial a  
point and terminal point are origin & point P respectively. It b c

is denoted by OP .
 
Magnitude of OP = | OP |= x 2 + y 2 + z 2
  
In the above figure a , b and c are collinear vectors
   
Types of Vectors where a and c are like vectors and a and b are unlike
vectors.
1. Zero or Null Vector: A vector whose magnitude is zero 5. Coplanar Vector: If the directed line segment of some

is called zero or null vector and it is denoted by 0. The given vectors lie in a plane then they are called coplanar
initial and terminal points of the directed line segment vectors. It should be noted that two vectors having the
representing zero vector are coincident and its direction same initial point are always coplanar but such three or
is arbitrary. more vectors may not be coplanar. [RC]
2. Unit Vector: A vector whose magnitude is unity is 6. Coinitial Vector: Vectors having same initial points are
called a unit vector. called coinitial vector.
 2 NTA CUET (UG) - Mathematics PW

7. Negative of a Vector: A vector is called negation of Properties of Vector Addition

a given vector if it has same magnitude but opposite
Vector addition has the following properties:
direction to the given vector.
1. Binary Operation: The sum of two vectors is always a
8. Reciprocal Vector: A vector which has the same
 vector.
direction as vector a but whose magnitude is the  
 2. Commutativity: For any two vector a and b,
reciprocal of the magnitude of a , is called the reciprocal
     
vector a and is denoted by a −1 . a+b = b+a
  1 aˆ   
Thus if a = α â, then a −1 = ⋅ aˆ =  2 3. Associativity: For any three vectors a, b and c
α a      
a + (b + c) = (a + b) + c

Note: A unit vector is self reciprocal.
4. Additive identity: For any vector a zero vector is the
additive identity.
    
Addition and Subtraction of Vector 0+a = a = a +0
5. Additive inverse: For every vector its negative vector

1. Triangle Law of Vector Addition: If two vectors are – a exists such that
represented by two consecutive sides of a triangle such     
a + (−a) = (−a) + a = 0
that the initial point of 2nd vector is same as terminal  
point of 1st vector then their sum is represented by the i.e. (– a ) is the additive inverse of the vector a .
 
third side of the triangle which is coinitial to a . This is Also if a = a1 i + a 2 j + a 3 kˆ
known as the triangle law of vector addition. 
C then −a =−a1 i − a 2 j − a 3 kˆ

   Subtraction of Vectors
c= a + b 
b
   
If a and b are two vectors, then their subtraction a − b is
     
defined as a − b = a + (−b) , where −b is the negative of b
 
a 
A B If a = a1ˆi + a 2 ˆj + a 3 kˆ and b = b1ˆi + b 2 ˆj + b3 kˆ
         
Thus, if=AB a,= BC b, and AC = c then a − b = (a1 − b1 )i + (a 2 − b 2 )ˆj + (a 3 − b3 )kˆ
     
then AB= + BC AC i.e.= a+b c Note: (i) Unlike the addition, commutativity &
associativeity does not hold for substraction of
2. Parallelogram Law of Vector Addition: If two
vectors.
coinitial vectors are represented by the two adjacent
sides of a parallelogram (as shown in fig.) then their
Useful Inequatlities
sum is represented by diagonal of the parallelogram,
which is coinitial to other vectors.
 Triangle inequality
B C    
(i) | a + b | ≤ | a | + | b |
       
b c= a + b (ii) | a − b | ≤ | a | + | b |
 Reverse triangle inequality
   
(i) | a + b | ≥ | a | − | b |
O  A    
a (ii) | a − b | ≥ | a | − | b |
     
Thus, if= OA a,= OB b, and OC = c
     
Then OA = + OB OC i.e.= a + b c. Vectors in terms of Position
3. Addition in Component form: If the vectors are Vectors of end Point
defined in terms of ˆi, ˆj and k̂ . 
 If AB be any given vector and suppose that the position
i.e. if a = a1ˆi + a 2 ˆj + a 3 kˆ and  
 vectors of initial point A and terminal point B are a and b
b = büˆi + b ˆj + b kˆ then their sum is defined as respectively,
      
a + b = (a + b )iˆ + (a + b )ˆj + (a + b )kˆ
1 1 2 2 3 3 then AB = OB − OA =− b a
Vector Algebra 3

i.e. AB = p.v. of point B – p.v. of point A
 Position Vector of a Dividing Point
AB
A (x1,y,1 z1) B(x2,y,2 z2)  
If a and b are the position vectors of two points A and B, then
  
a the position vector c of apoint P dividing AB in the ratio m :
b 
 mb + na
O n internallyis then c =
 m+n
= AB = ( x2 − x1 )iˆ + ( y2 − y1 ) ˆj + ( z2 − z1 )kˆ B

Distance between two Points n

Let A and B be two given points whose coordinate are  P

b
respectively (x1,y1, z1) and (x2, y2, z2)
 
 c m
If a and b are p.v. of A and B relative to point O, then

a = x1i + y1 j + z1kˆ

b = x 2 i + y 2 j + z 2 kˆ A
     O 
a
Now AB = OB − OA =− b a
= (x 2 − x1 )i + (y 2 − y1 )ˆj + (z 2 − z1 )kˆ

Particular Case

Distance between the points = magnitude of AB  
a+b
(i) Position vector of the mid point of AB is
= (x 2 − x1 ) 2 + (y 2 − y1 ) 2 + (z 2 − z1 ) 2 2
(ii) Any vector along the internal bisector of ∠AOB is
 
given by λ(a + b) where λ is any scalar.
Multiplication of a Vector by Note: (i) If the point P divides AB in the ratio m : n
a Scalar 
externally,
 and c be the position vector of P then

   mb − na
If a is a vector and m is a scalar (i.e. a real number) then m a c=
 m−n
is a vector whose magnitude is m times that of a and whose   
 (ii) If a, b, c are position vectors of vertices of a
direction is the same as that of a , if m is positive and opposite   
 a+b+c
to that of a , if m is negative, triangle, then p.v. of its centroid is
 3
Again if a = a1i + a 2 ˆj + a 3 kˆ then (iii) If a,b,c,d are position vectors of vertices
 of a tetrahedron, then p.v. of its centroid is
ma = (ma1 )iˆ + (ma 2 )ˆj + (ma 3 )kˆ   
   a+b+c+d
∴ magnitude of m a =| m a | = m | a |
4
Note:
(i) The multiplication of a vector by a scalar is also named Product of Vectors
as ‘scalar multiplication’.
(ii) From the definition of Scalar multiplication it is obvious Product of two vectors is done by two methods when the
    product of two vectors results in a scalar quantity then it is
to note that a || b ⇒ a =mb, where m is some suitable
called scalar product. It is also called as dot product because
scalar. this product is represented by putting a dot.
Properties When the product of two vectors results in a vector quantity
  then this product is called. Vector Product. This product is
If a and b are any two vectors and m, n are any scalar then represented by × sign so that it is also called as cross product.
  
(i) m( a ) = ( a ) m = m a (commutativity)
   Scalar or Dot Product of Two Vectors
(ii) m (n a ) = n (m a ) = (mn) a (Associativity)  
   If a and b are two non zero vectors and q be the angle between
(iii) (m + n)a = ma + na  them, then their scalar product (or dot product) is denoted as
     (Distributivity)  
(iv) m(a + b) = ma + mb  a ⋅ b and it is defined by
 4 NTA CUET (UG) - Mathematics PW
    (iii) With the help of the above cases, we get the following
a ⋅ b | a || b | cos θ ,
= ≤θ ≤ π
important results:
B
Y

 j
b
k X
q i
O
Z
 A
a
(a) ˆi ⋅ ˆi = ˆj ⋅ ˆj = kˆ ⋅ kˆ = 1
 
Note: (i) a ⋅ b ∈ R (b) ˆi ⋅ ˆj = ˆj⋅ kˆ = kˆ ⋅ ˆj = 0
       
(ii) a ⋅ b ≤ | a || b | (iv) If a and b are unit vectors, then a ⋅ b= cos θ
   
(iii) If q = 0 then a ⋅ b =| a || b | Properties of Scalar Product
π     
(iv) If q = then a ⋅ b = 0 If a, b, c are any vectors and m, n any scalars then
  2      
(v) a ⋅ b > 0 ⇒ angle between a and b is acute (i) a ⋅ b = b ⋅ a (Commutativity)
         
(vi) a ⋅ b < 0 ⇒ angle between a and b is obtuse. (ii) (m a) ⋅ b = a ⋅ (m b) = m(a.b)
   
(iii) (m a) ⋅ (n b)= (mn)(a ⋅ b)
Geometrical Interpretation
      
Geometrically, the scalar product of two vectors is equal to the (iv) a ⋅ (b + c) = a ⋅ b + a ⋅ c (Distributivity)
     
product of the magnitude of one and the projection of second (v) a ⋅ b = a ⋅ c ⇒ b = c (Right Cancellation)
in the direction of first vector i.e.       
Infact a ⋅ b = a ⋅ c ⇒ a ⋅ (b − c) = 0
            
a=⋅ b a(b cos θ) = | a | (projection of b in the direction of a ) ⇒= a 0 or= b c or a ⊥ (b − c)
       
Similarly= a.b b(a cos θ) = | b | (projection of a in the (vi) (a ⋅ b) ⋅ c is meaningless
 (vii) Scalar product is not binary operation.
direction of b )
B Note:
  
(a) (a ⋅ b) ⋅ b is not defined
  2  2    2
L (b) (a + b) = | a | + 2a ⋅ b + | b |
   2  2    2
b (c) (a − b)= | a | − 2a ⋅ b + | b |
     2  2
(d) (a + b) ⋅ (a − b) = | a | − | b |
      
O M a
A (e) | a + b |= | a | + | b |⇒ a || b
       
  a⋅b (f) | a + b |2= | a |2 + | b |2 ⇒ a ⊥ b
Here projection of b on a =  and      
|a| (g) | a + b |= | a − b | ⇒ a ⊥ b
 
  a ⋅b
Projection of a on b =  Scalar Product in Terms of Components
|b|  
Let a and b be two vectors such that
Scalar Product in Particular Cases  
a = a1ˆi + a 2 j + a 3 k and b = b1ˆi + b 2 j + b3 k
   
(i) If a and b are like vectors, then q = 0 so Then a ⋅ b= a1 b1 + a 2 b 2 + a 3 b3
    
a ⋅=b | a ||= b | a b i.e. scalar product of two like vectors
is equal to the product of their modulii Angle Between Two Vectors
  
 (i) If a and b be two vectors and q be the angle between
(ii) If a and b are unlike vectors then q = p so
         
a ⋅ b =| a || b | cos π = − | a || b | a ⋅b −1  a ⋅ b 
them, then cos θ =   ⇒ θ =cos    
    | a || b |
But its converse may not be true i.e. a ⋅ b = 0 ⇒ / a⊥b  | a || b | 
   
But if a and b are non zero vectors, then (ii) If a = a1ˆi + a 2 j + a 3 k and b = b1ˆi + b 2 j + b3 k then
   
a ⋅b = 0 ⇒ a ⊥ b a1b1 + a 2 b 2 + a 3 b3

cos θ =
     a + a 22 + a 32 b12 + b 22 + b32
2
Thus a ≠ 0, b ≠ 0, a ⋅ b = 0, ⇒ a ⊥ b 1
Vector Algebra 5
   
Note: If a and b are perpendicular to each other then a1 b1 + If n̂ is the unit vector perpendicular to the plane of a and b,
a2 b2 + a3 b3 = 0. then
 
a×b
Component of B Along & Perpendicular to A n̂ =  
| a×b |
B
Vector Product in Particular Cases
   
r
b ( )
(i) If θ =0 i.e. a || b then a × b =0
π      
r
(ii)=If θ
2
( ) | a || b | nˆ
i.e. a ⊥ b then a × b =
 a (iii) If ˆi, ˆj, kˆ be three mutually perpendicular unit vectors,
O M A then
 
(i) Component along a = OM ĵ
= OM = aˆ (b cos θ) aˆ
  k̂
(ab cos θ) (a ⋅ b) 
= = â ⋅a
a a2
 
(ii) Component perpendicular to a = MB î
   
= MO + OB = OB − OM (a) ˆi × ˆi = ˆj × ˆj = kˆ × kˆ = 0

 (a ⋅ b)  (b) ˆi × ˆj =k,ˆ ˆi × kˆ =−ˆj, kˆ × ˆi =ˆj
= b − 2 ⋅a
a (c) ˆj × ˆi =−k,
ˆ kˆ × ˆj =−ˆi, ˆi × kˆ =−ˆj
Work Done by The Force
 Properties of Vector Product
If a constant force F acting on a particle displaces it from   
  If a , b, c are any vectors and m,n any scalars then
point A to B, then work done by the force W = F ⋅ d
         
(i) a × b ≠ b × a (Non- commutativity) but a × b =−( b × a)
(where d = AB )    
and | a × b | =|b × a |
     
(ii) (ma) × b = a × (m b) = m(a × b)
Vector or Cross Product of    
Two Vectors (iii) (ma) × (nb)= (mn)(a × b)
     
 (iv) a × (b × c) ≠ (a × b) × c
       
If a and b be two vectors and q(0 ≤ q ≤ p) be the angle (v) a × (b × c) = (a × b) × (a × c) (Distributivity)
between them, then their vector (or cross) product is denoted      
  (vi) a × b = a × c ⇒ / b = c Infact
by a × b and it is defined by
      
    a × b = a × c ⇒ a × (b − c) = 0
∴= a × b | a || b |sin θ nˆ     
 
 ⇒= a 0 or= b c or a || (b − c)
Where n̂ is a unit vector perpendicular to the plane of a and b
 
such that a , b and nˆ form a right handed system. Geometrical Interpretation of Vector Product

Vector Product in Terms of Components D C

     
If a = a1ˆi + a 2 ˆj + a 3 kˆ and b = b1ˆi + b 2 ˆj + b3 kˆ then a × b  d2 d1
b
ˆi ˆj kˆ
 
a × b = a1 a2 a3 A  B
a
b1 b2 b3
 
If a and b be the two adjcent side of a parallelogram ABCD
= (a2 b3– a3b2) î + (a3b1 – a1b3) ĵ + (a1b2 – a2b1) k̂
then
Angle between two Vectors  
(i) Area of parallelogram= | a × b |
  
 |a×b| 1  
If q is the angle between a and b, then sin θ =   (ii) Area of parallelogram
= (d1 × d 2 )
| a || b | 2
 6 NTA CUET (UG) - Mathematics PW
 
where d1 & d 2 are diagonal vector of parallelogram a1 a2 a3
ABCD. 

[a b c] = b1 b2 b3
Area of a Triangle c1 c2 c3
  
C (iii) For any three vectors a, b and c
       
(a) [a + b b + c c + a] = 2[a b c]
   
(b) [a − b b − c c − a] = 0
     
B (c) [a × b b × c c × a] =[a b c]2
A
1   Properties of Scalar Triple Product [RC]
(i) Area of ∆ABC = | AB × AC |
   2 (i) The position of (.) and (×) can be interchanged, i.e.,
(ii) If a, b, c are position vectors of vertices of a DABC            
then its a ⋅ (b × c) = (a × b) ⋅ c but also (a × b) ⋅ c = c ⋅ (a × b)
    
1       So [a= b c] [b= c a] [c a b]
Area of parallelogram Area= | (a × b) + (b × c) + (c × a) |
2   Therefore if we don’t change the cyclic order of a, b and
Note: Three points with position vectors a, b, c are collinear if c then the value of scalar triple product is not changed
     
(a × b) + (b × c) + (c × a) = 0 by interchanging dot and cross.
(ii) If the cyclic order of vectors is changed, then sign of
Moment of a Force      
scalar triple product is changed i.e. a ⋅ [b × c] =−a ⋅ (c × b)
The moment of the force F acting at a point A about O is given  
or [a b c] = −[a c b]
by
     from (i) and (ii) we have
Moment of F = OA × F = r × F      
[a= b c] [b= c a] [c a b] = −[a c b] = −[b a c] = −[c b a]
(iii) The scalar triple product of three vectors when two of
Scalar Triple Products [RC]  
 them are equal or parallel, is zero i.e. [a= b b] [ =
a b a] 0
   (iv) The scalar triple product of three mutually perpendicular
If a, b, c are three vectors, then their scalar triple product is
  
defined as the dot product of two vectors a and b × c. It is unit vectors is ±1 Thus [iˆ ˆjk]
ˆ = 1, [iˆ kˆ ˆj] = −1
   
    
generally denoted by a ⋅ (b × c) or[a b c]. Which is read as box (v) If two of the three vectors a, b, c are parallel then
   
 
product of a, b, c. Similarly other scalar triple products can be [a b c] = 0
defined as    
      (vi) a, b, c are three coplanar vectors if [a b c] = 0 i.e. the
(b × c) ⋅ a,(c × a) ⋅ b necessary and sufficient condition for three non-zero
Note: Scalar triple product always results in a scalar quantity.  
collinear vectors to be coplanar is [a b c] = 0
   
Geometrical Interpretation [RC] (vii) For any vectors a, b, c, d
     
The scalar triple product of three vectors is equal to the [a + b c d] = [a c d] + [b c d]
volume of the parallelopiped whose three co-terminous edges
are represented by the given vector. Volume of Tetrahedron [RC]
      
Therefore (a × b)=⋅ c [a b=
c] Volume of the parallelopiped (i) If a, b, c are position vectors of vertices A, B and C
   with respect to O, then volume of tetrahedron OABC
whose three coterminous edges are a, b and c
Formula for scalar Triple Product: 1 
= [a b c]
  6   
(i) If a = a1 î + a2 ĵ + a3 k̂ , b = b1 î + b2 ĵ + b3 k̂ and (ii) If a, b, c,d are position vectors of vertices A,B,C,D of a
a1 a 2 a 3 tetrahedron ABCD, then
   1     
c = c1 î + c2 ĵ + c3 k̂ , =then [a b c] = b1 b 2 b3 [lmn]
 6  AB AC AD 
c1 c 2 c3 
Its volume =  or
 
(ii) a = a1ˆi + a 2 ˆj + a 3 k,
ˆ b = b ˆi + b ˆj + b kˆ and 1
 1 2 3
 [b − a c − a d − a ]
c = c1ˆi + c 2 ˆj + c3 k,
ˆ then
6