JEE | NEET | Class 8 - 10 Download eSaral App

Revision Notes
Class 12 Maths
Chapter 10 – Vector Algebra

Vector
Vector quantities are those quantities that have magnitude and direction. It is
generally represented by a directed line segment. We represent a vector as AB ,
where initial point of vector is denoted by A and the terminal point by B . The
magnitude of vector is expressed as AB .

Position Vector
Let us denote the origin as O such that this is a fixed point. There is a point, say
P at a distance from O . Now, the position vector of a point P is given by the
vector OP .
The next case is when there are two vectors, a and b which represent the position
vectors of two points A and B . Then we can write the vector AB=b − a or the
position vector of B − the position vector of A .

Types of vectors
1. Zero Vector – It has zero magnitude. This means that vector has the same
initial and terminal point. It is denoted by O . The direction of zero vector
is indeterminate.

2. Unit Vector – It has unit magnitude. Unit vector in direction of a vector

a
a is denoted by a and symbolically as a= .
a

3. Co-initial Vectors - Two or more vectors are said to be co-initial if they

have the same initial point.

4. Equal Vectors - Two vectors are said to be equal if they have the same
magnitude and direction. They represent the same physical quantity.

5. Collinear Vectors – Two or more vectors are said to be collinear if they

are parallel to the same line irrespective of their direction. For this reason,
they are also called parallel vectors. We have two sub-categories – like
1
Rev. Notes Class 12 Maths Chapter- 10 www.esaral.com
 JEE | NEET | Class 8 - 10 Download eSaral App

vectors (same direction) and unlike vectors (different directions). We can

represent it mathematically by taking two non-zero vectors a and b . They
are collinear if and only, if a=Kb , where K  R-0 .

6. Coplanar Vectors – Those vectors which lie on the same plane and they
are all parallel to the same plane. We must remember that two vectors are
always coplanar.

7. Negative Vector – A vector which has same magnitude but opposite

direction to another vector is called negative of that vector.

Addition of vectors
1. Triangle Law – Consider a triangle ABC . Let the sum of two vectors
a and b be represented by c . The position vectors are represented by
AB , BC and AC .

Triangle law of vector addition states that when two vectors are represented as
two sides of the triangle with the order of magnitude and direction, then the third
side of the triangle represents the magnitude and direction of the resultant vector.
So, we can write that AC = AB + BC .

2. Parallelogram Law – Consider a parallelogram ABCD . Let the sum of

two vectors a and b be represented by c . The position vectors are
represented as
a =AB = DC
b =AD = BC
a + b=AC

Rev. Notes Class 12 Maths Chapter- 10 www.esaral.com 2

 JEE | NEET | Class 8 - 10 Download eSaral App

According to the parallelogram law of vector addition if two vectors act along
two adjacent sides of a parallelogram (having magnitude equal to the length of
the sides) both pointing away from the common vertex, then the resultant is
represented by the diagonal of the parallelogram passing through the same
common vertex and in the same sense as the two vectors.
The sum is
AB + BC = AC
AB + AD = AC

3. Properties of vector addition

a) Commutative property - a+b=b+a

( )
b) Associative property - a+b +c=a+ b+c ( )
c) Zero is the additive identity - a+0=a=0+a

( )
d) a+ -a =0= -a +a ( )
Multiplication of a vector by a scalar
If a is a vector and m is a scalar, then their product is m a . The magnitude would
be m times the magnitude of a . This is called scalar multiplication. If a and
b are vectors and m and n are scalars, then

() ()
a) m a = a m=ma
b) m ( na ) =n ( ma ) = ( mn ) a
c) ( m+n ) a=ma+na
( )
d) m a+b =ma+mb
Rev. Notes Class 12 Maths Chapter- 10 www.esaral.com 3
 JEE | NEET | Class 8 - 10 Download eSaral App

Component form of vectors

● We have to consider three axis - x, y,z and a point in the coordinate axis.
So, the position vector for such a point would be written as
OP = xiˆ + yjˆ + zkˆ . This is the component form of vector.
● The scalar components are x, y,z and the vector components are
ˆ yj,zk
xi, ˆ ˆ.
● Consider two vectors as A = aiˆ + bjˆ + ckˆ and B = piˆ + qjˆ + rkˆ , then
a) Sum is given by A + B = (a + p)iˆ + (b + q)ˆj + (c + r)kˆ .
b) Difference is given by A − B = (a − p)iˆ + (b − q)ˆj + (c − r)kˆ .
c) Multiplication by a scalar m is given by mA = maiˆ + mbjˆ + mckˆ .
d) The vectors are equal if a = p,b = q,c = r .

Test for collinearity

Three points A,B,C with position vectors a,b,c respectively are collinear, if and
only if there exist scalar x,y,z not all zero simultaneously such that; xa+yb+zc=0
, where x+y+z=0 .

Test for coplanar points

Four points A,B,C,D with position vectors a,b,c,d respectively are coplanar if
and only if there exist scalars x,y,z,w not all zero simultaneously such that;
xa+yb+zc+wd=0 , where x+y+z+w=0 .

Section formula
a) Let a and b be the position vectors of two points A and B . A point R
with position vector as r divides AB such that mRB = nAR and this
denotes that AB is divided internally in the ratio m:n is given by
mb + na
r= .
m+n
b) Let a and b be the position vectors of two points A and B . A point R
with position vector as r divides AB such that mRB = nAR and this
denotes that AB is divided externally in the ratio m:n is given by

Rev. Notes Class 12 Maths Chapter- 10 www.esaral.com 4

 JEE | NEET | Class 8 - 10 Download eSaral App

mb − na
r= .
m−n
c) Now if the ratio is 1:1, then we can obtain the position vector of the
a+b
midpoint as .
2

Magnitude of vector
a) For a vector A = aiˆ + bjˆ + ckˆ , magnitude is A = a 2 + b 2 + c 2 .
b) For vector AB with A = aiˆ + bjˆ + ckˆ and B = piˆ + qjˆ + rkˆ , the magnitude is
AB = (p − a) 2 + (q − b) 2 + (r − c) 2 .

Product of vectors
1. Scalar Product
● It is also called dot product. For two vectors a and b , the dot product can
be represented as a.b and it is defined as a  b= a b cosθ;(0  θ  π) .
a.b
● From this, we can find the angle between vectors as cos  = .
a b
● We have the below possibilities:
a) If θ is acute, then a  b>0 .
b) If θ is obtuse, then a  b<0 .
c) If θ is zero, then a  b = a b .
d) If θ is  , then a  b = − a b .
● If vectors a and b are non-zero and a.b = 0 , then it is the condition for
them to be perpendicular vectors.
● Considering component form and above point, we get results as

a) i  i= j  j=k  k=1

b) i  j= j  k=k  i=0

● If a=a1 i+a 2 j+a 3 k and b=b1 i+b 2 j+b3 k then a  b=a1b1 + a 2b 2 + a 3b3 .
● Properties of scalar product

Rev. Notes Class 12 Maths Chapter- 10 www.esaral.com 5

 JEE | NEET | Class 8 - 10 Download eSaral App

2 2
a) a  a= a =a ,a  b=b  a (Commutative)

( )
b) a  b+c =a  b+a  c (Distributive)

c) ( ma )  b=a  ( mb ) =m ( a  b ) (Associative), where m is scalar.

ab
● Projection of vector a on b= .
b
● Maximum value of a  b= a b
● Minimum value of a  b=- a b
● A vector in the direction of the bisector of the angle between two vectors
a b
a and b is + .
a b

● Hence bisector of the angle between the two vectors a and b is λ a + b , ( )

where λ  R + .
( )
● Bisector of the exterior angle between a and b is λ a-b λ  R-0 .

2. Vector Product
● It is also called cross product. For two vectors a and b , the vector product
is represented as a  b and is defined by a×b= a b sinθn , where  is the
angle between them and n is the unit vector perpendicular to both a and
b such that a , b and n form a right handed screw system.

ab
● From this, we can write the angle between vectors as sin  = .
a b

● If vectors a and b are non-zero and a  b = 0 , then it is the condition for

them to be parallel vectors.
● Considering component form and above point, we get results as

a) i×i= j×j=k×k=0

Rev. Notes Class 12 Maths Chapter- 10 www.esaral.com 6

 JEE | NEET | Class 8 - 10 Download eSaral App

b) i×j=k, j×k=i,k×i= j
i j k
● If a=a1 i+a 2 j+a 3 k and b=b1 i+b 2 j+b3 k then a×b= a1 a 2 a.
b1 b 2 b
● Geometrically, we can define a×b = area of the parallelogram whose two
adjacent sides are represented by a and b .
● Properties of vector product
a) a×b  b×a (Not Commutative)

b) ( ma )×b=a×( mb ) =m ( a×b ) (Associative) where m is scalar.

c) a×( b+c ) = ( a×b ) + ( a×c ) (Distributive)

● Unit vector perpendicular to the plane of a and b is

a×b
n=±
a×b

● A vector of magnitude 'r' and perpendicular to the plane of a and b is

±
( )
r a×b
a×b

● If a,b and c are the position vectors of vertices A,B and C of a triangle,
then the vector area of triangle is given by
1
ABC= a×b + b×c + c×a 
2
The points A,B and C are collinear if a×b+b×c+c×a=0 .

● Area of quadrilateral whose diagonal vectors are d1 and d 2 is given by

1
d1  d 2 .
2

Rev. Notes Class 12 Maths Chapter- 10 www.esaral.com 7

 JEE | NEET | Class 8 - 10 Download eSaral App

Scalar Triple Product

● The scalar triple product of three vectors a,b and c is defined as a. b  c ( )
and can be represented as  a b c  . It is also referred to as box product.
● Geometrically, it represents the volume of the parallelepiped whose three
coterminous edges are represented by a,b and c . So V= a b c  .
● Scalar triple product is cyclic, i.e. the order of vectors can be interchanged
in a cyclic manner as shown below,
( ) ( )
a  b  c = a  b  c or a b c  =  b c a  = c a b 
a  ( b  c ) = -a  ( c  b ) or a b c  = - a c b 
● If a=a1 i+a 2 j+a 3 k ; b=b1 i+b 2 j+b3 k and c=c1 i+c 2 j+c3 k then
a1 a 2 a
a b c  = b1 b 2 b
 
c1 c2 c

● Scalar product of three vectors, two of which are equal or parallel is 0 .

● Vectors a, b,c are coplanar if a b c  =0 .

Rev. Notes Class 12 Maths Chapter- 10 www.esaral.com 8