Question
State and prove associative law for vector addition.
Solution
The associative law of vector states that the vector addition is same, in whatever grouping they are added.
Proof:
Let there be three vectors
. These three vectors, both magnitude and direction are represented by three vectors 
of polygon OLKG.
Now, using polygon law of vector addition, the resultant of the vector will be represented by the closing side of the polygon
.
Join O with K, L and G.
Now, applying triangle law to
, we have
Now, in
, we have
Now, applying the triangle law of vector addition to
we have
For triangle OLG, we get
Now, from equations (1) and (2), we get
Therefore, the vector addition is assosciative.
Proof:
Let there be three vectors
. These three vectors, both magnitude and direction are represented by three vectors of polygon OLKG. Now, using polygon law of vector addition, the resultant of the vector will be represented by the closing side of the polygon
. Join O with K, L and G.
Now, applying triangle law to
, we have Now, in
, we haveNow, applying the triangle law of vector addition to
we haveFor triangle OLG, we get
Now, from equations (1) and (2), we get
Therefore, the vector addition is assosciative.
