CHAPTER 2 - Addition of Vectors

You will be able understand the discussion that follows very clearly only if you try to visualise everything physically. Everything about vectors will then automatically fall in place in your mind.

(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  and walks first along  and then along  will reach the point . Thus, the combined effect of  and  is to take the person from  to , 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  and .

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.