Chapter 2 : Algebra of Vectors
Topics covered in this snack-sized chapter:
Two or more vectors may be added together to produce their addition.
If two vectors have the same direction, their resultant has a magnitude equal to the sum of their magnitudes and will also have the same direction.
Similarly orientated vectors can be subtracted in the same manner.
In general, since vectors may have any direction, we must use one of two methods for adding vectors.
If two vectors 
and 
are represented in magnitude and direction by the two adjacent sides of a parallelogram then their sum 
is represented by the diagonal of the parallelogram which is co-initial with the given vectors.
In the parallelogram method for vector addition, the vectors are translated, (that is, moved) to a common origin and the parallelogram constructed as follows:
Step 1:
Step 2:
Step 3:
Symbolically, we have
or 
We have
Therefore, in triangle OPR,
Thus, it follows that if two vectors are represented in magnitude and direction by the two sides of a triangle taken in the same order, then their sum is represented by the third side taken in the reverse order.
This is called the triangle law of addition of vectors.
Using triangle law of addition of vectors in 
we have:
and 
Example: Find the sum of vectors 
and 
?
Solution: We have,
Commutative:
For any two vectors and
, we have,
Associative:
For any three vectors, 
we have,
Existence of additive identity:
For every vector
, we have,
,
- Where,
is the null vector.
Existence of additive inverse:
For every vector, there corresponds a vector 
such that,
If and are two vectors, then the subtraction of from is defined as the vector sum of and 
and it is denoted by 
i.e. 
.
Thus, to subtract vector from vector, we reverse the direction of vector and add it to vector as:
Example: If 
and
then find
?
Solution: We have,
