(A) ADDITION OF VECTORS : TRIANGLE / PARALLELOGRAM LAW
Most of you will already be very familiar with how to add vectors, from your study of physics.
Consider two vectors  and  which we wish to add. Let
 = +
Thus, should have the same effect as and combined. To find the combined effect of and ,we place the initial point of on the end-point of (or vice-versa):
A person who starts at point A and walks first along and then along will reach the point C. Thus, the combined effect of and is to take the person from A to C, i.e, + = should be the vector  :
In general, we see that to add two vectors, say and , we place the initial point of one of them, say at the end-point of the other, i.e., . The vector + is then the vector joining the tip of to the end-point of . This is the triangle law of vector addition. and can equivalently be added using the parallelogram law; we make the two vectors co-initial and complete the parallelogram with these two vectors as its sides:
The vector  then gives us the sum of a and b
Note that the triangle and the parallelogram law are entirely equivalent; they are two slightly different forms of the same fundamental principle.
We note the following straightforward facts about addition.
(a) Existence of identity: For any vector ,
+  =
so that vector is the additive identity.
(b) Existence of inverse: For any vector ,
+ (-) =
and thus an additive inverse exists for every vector.
(c) Commutativity: Addition is commutative; for any two arbitrary vectors and
+ = +
(d) Associativity: Addition is associative; for any three arbitrary vectors , and
+ ( + )= ( + )+
i.e, the order of addition does not matter.
Verify this explicitly by drawing a vector diagram and using the triangle / parallelogram
law of addition.
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