7.2 Algebra of Vectors
(1) Addition of two Vectors

Triangle Law : The sum or resultant of two vectors
a and b is the vector formed by
placing the initial point of b on the terminal point of
a and then joining the initial point of
a to the terminal point of b .
This sum is written as r = a + b
This law is also known as Tail  Tip
rule
Parallelogram
Law : It states that the sum or resultant of
a and b is R is the diagonal of
the parallelogram for which a and b are adjacent sides. All vectors
a, b and R are concurrent as shown
in figure 4.
Vector addition is commutative i.e.
a + b = b+ c Extension to sums of two or more vectors
is immediate.
Vector addition is associative. Observe the adjoining fig.5 we have using tailtip rule, for triangle for
triangle ABC, for
triangle OBC, c
i.e. r = ( a + b)
+ c . Also, for triangle OAC ,
i.e
r = a + (b + c).
Thus , r = ( a + b ) + c = a + (b + c).
2) Subtraction of a Vector : It is accomplished by adding the negative of vectors i.e. a  b = a + (  b). Note also that  ( a + b ) =  a  b .
3) Composition of Vectors :
It is the process of determining the resultant of a system of vectors.
For this we have to draw a vector polygon, placing tail of each vector at the tip of the
preceding vector. Then draw a vector from the tail of the first vector to tip of the last vector in the system.
Later on we will show that not all vector systems reduce to a single vector. Also note that the order of
vectors are drawn is immaterial.
Free Vector : A vector which can be shifted along its line of action or parallel to itself and whose initial point can be anywhere, is called a free vector.
Unless stated, we always refer a vector as a free vector .
Localized Vector
: It is also known as a sliding vector. It can be shifted only along its line of action.
By the principle of transmissibility the external effects of this vector remains the same.
Bounded Vector : It is called as a fixed vector. It must remain at the same point of action
or application.
Negative of a Vector
: The negative of a vector u is the vector
u which has the same direction and inclination but is of the opposite sense (of direction).
* The resultant of a
system of vectors is the least number of vectors that will replace
the given system.
Position Vector of a Point
: If A and B are any two points then is called the position
vector of point B with reference to point A.
Usually we fix a point O and define the position vector of points with reference to O which is called the
origin of reference.
We shall use a, b, c etc. to denote position vectors of points A, B, C etc. respectively, relative to origin
O.
Unit Vector : A vector in a given direction having unit magnitude is called a unit vector in
that direction .
It is usually denoted by etc ( Read as n Cap). Also, note that if
U is given vector then the unit vector in the direction (or along or parallel ) to
u is given by
