Summary of Class 12 Maths Vector Algebra
Vector
A quantity that has magnitude as well as direction is called a vector.
It is denoted as 
or simply
as 
. A is called the initial
point and B is called the terminal point of vector .
Magnitude of a Vector
The distance between initial and terminal points of a vector 
or 
is called the magnitude
of or and it is denoted by
| |
or || or a.
Note: Since, the distance is never negative, the notation || < 0 has no
meaning.
Position Vector
Let O(0, 0, 0) be the origin and P be a point in
space having coordinates (x, y, z) with respect to the origin O. Then, the
vector 𝑂𝑃 or 
is called the
position vector of the point P with respect to O. Its magnitude is given by
Direction Cosines
Let
α, β and γ be the angles made by the vector
or 
with
the positive directions of the coordinate axes OX, OY and OZ, respectively. Then
cos α, cos β and cos γ are known as the direction cosines of or and are generally denoted by the letters l, m and n,
respectively.
That is, l = cos α, m = cos β, n
= cos γ
If l, m and n be the direction
cosines of ,
then l2 + m2 + n2 =
1.
Direction Ratios
From the above figure, we can write that cos α = x/r,
cos β = y/r and cos γ = z/r.
Thus, the coordinates of the point P may also be
expressed as (lr, mr, nr). The numbers lr, mr and
nr, proportional to the direction cosines are called as direction
ratios of vector , and
denoted as a, b and c, respectively.
Types of Vectors
Zero Vector: A vector whose
initial and terminal points coincide, is called a zero vector (or null vector).
Unit Vector: A vector of
unit length is called unit vector. The unit vector in the direction of is 𝑎̂ =
Coinitial
Vectors: Two or more vectors having the same initial
point are called coinitial vectors.
Collinear Vectors: Two or more vectors are said to be collinear, if they are parallel
to the same line, irrespective of their magnitudes and directions, e.g. and 
are
collinear, when 
= ±𝜆 or || = 𝜆||. If
two vectors are parallel or collinear, one can be expressed as a scalar
multiple of the other.
Equal Vectors: Two vectors are said
to be equal, if they have the same magnitude and the same direction. If = 
, then | I = ||.
Negative Vector: A vector having the
same magnitude but opposite in direction of the given vector, is called as its
negative vector. For example, vector 
is the negative of the vector 
and written as = – .
Addition of Vectors
Triangle Law of Vector Addition: If two vectors are represented along two sides of a triangle taken in order, then their resultant is represented by the third side taken in opposite direction. In ∆ABC, by triangle law of vector addition, we have
+ 
= 
.
Note: The vector sum of three sides
of a triangle taken in order is 
.
Parallelogram Law of Vector Addition: If two vectors are represented along the two
adjacent sides of a parallelogram, then their resultant is represented by the
diagonal of the parallelogram through their initial point.
If OA and OB are the sides of the
parallelogram, then OA + OB = OC.
Properties of Vector Addition
Property 1: For any two vectors and , + = + (Commutative property)
Property 2: For any three vectors , and 
, ( + ) + = + ( + ) (Associative property)
Property 3: for any vector , we have + = + =
Here, the zero vector is called the additive identity for the
vector addition.
Multiplication of a Vector by a Scalar
Let be a given vector and λ be a scalar. Then, the
product of the vector by the scalar λ, denoted as λ, is called the
multiplication of vector by the scalar λ.
Components of a Vector
Let the position vector of P with reference to O be 
. Let 
= = 𝑥𝑖̂ + 𝑦𝑗̂ + 𝑧𝑘̂. Here, x, y and z are called the scalar components
of and 𝑥𝑖̂, 𝑦𝑗̂ and 𝑧𝑘̂ are called the
vector components of along the respective
axes.
Vector Joining of Two Points
If P1(x1, y1, z1)
and P2(x2, y2, z2) are any two
points, then the vector joining P1 and
P2 is the vector 
.
Section Formula
The position vector 
of point R, which
divides the line segment joining the points A and B with position vectors and , respectively,
internally in the ratio m : n is given by
If P divides AB externally in the ratio m : n, then
Note: If R is the midpoint of AB, then OR = ( + ) / 2
Scalar (or Dot) Product of Two Vectors
If θ is the angle between two vectors and , then the scalar or
dot product of and , denoted by .
, is given by ⋅ = ∣∣∣∣∣ 𝑐𝑜𝑠 𝜃, where 0 ≤ θ ≤ π.
Note:
(i) ⋅ is a real number.
(ii) If either = or = , then θ is not
defined.
Properties of dot product of two vectors 
and 
are
as follows:
Vector (or Cross) Product of Two Vectors
If θ is the angle between two non-zero vectors and , then the vector
or cross product of the vectors, denoted by × , is
given by
where, 𝑛̂ is a
unit vector perpendicular to both and .
Note: × is
a vector quantity, whose magnitude is ∣ × ∣ = ∣∣|| 𝑠𝑖𝑛 𝜃
Properties of cross product of two vectors 𝑎⃗ and 𝑏⃗ are as follows:
