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State and prove associative law for vector addition.

The associative law of vector states that the vector addition is same, in whatever grouping they are added.

left parenthesis A with rightwards harpoon with barb upwards on top plus B with rightwards harpoon with barb upwards on top right parenthesis plus C with rightwards harpoon with barb upwards on top space equals space A with rightwards harpoon with barb upwards on top plus left parenthesis B with rightwards harpoon with barb upwards on top plus C with rightwards harpoon with barb upwards on top right parenthesis

Proof: 


Let there be three vectors straight A with rightwards harpoon with barb upwards on top comma space B with rightwards harpoon with barb upwards on top space a n d space C with rightwards harpoon with barb upwards on top. These three vectors, both magnitude and direction are represented by three vectors stack OL comma with rightwards harpoon with barb upwards on top space stack L K with rightwards harpoon with barb upwards on top space a n d space stack K G with rightwards harpoon with barb upwards on top space 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 OG with rightwards harpoon with barb upwards on top
Join O with K, L and G. 

Now, applying triangle law to triangle OLK, we have

               OK with rightwards harpoon with barb upwards on top space equals space stack O L with rightwards harpoon with barb upwards on top plus stack L K with rightwards harpoon with barb upwards on top space equals space A with rightwards harpoon with barb upwards on top plus B with rightwards harpoon with barb upwards on top 

Now, in increment OKG, we have

OG with rightwards harpoon with barb upwards on top space equals space stack O K with rightwards harpoon with barb upwards on top plus stack K G with rightwards harpoon with barb upwards on top space equals space left parenthesis A with rightwards harpoon with barb upwards on top plus B with rightwards harpoon with barb upwards on top right parenthesis space plus space C with rightwards harpoon with barb upwards on top space space space space space space space space space space space space space space space... space left parenthesis 1 right parenthesis

Now, applying the triangle law of vector addition to increment LKG comma spacewe have

LG with rightwards harpoon with barb upwards on top space equals space stack L K with rightwards harpoon with barb upwards on top plus stack K G with rightwards harpoon with barb upwards on top space equals space B with rightwards harpoon with barb upwards on top space plus space C with rightwards harpoon with barb upwards on top

For triangle OLG, we get

OG with rightwards harpoon with barb upwards on top space equals space stack O L with rightwards harpoon with barb upwards on top plus stack L G with rightwards harpoon with barb upwards on top space equals space A with rightwards harpoon with barb upwards on top space plus space left parenthesis B with rightwards harpoon with barb upwards on top plus C with rightwards harpoon with barb upwards on top right parenthesis space space space space space space space space space space space... space left parenthesis 2 right parenthesis

Now, from equations (1) and (2), we get

left parenthesis straight A with rightwards harpoon with barb upwards on top plus B with rightwards harpoon with barb upwards on top right parenthesis space plus space C with rightwards harpoon with barb upwards on top space equals space A with rightwards harpoon with barb upwards on top space plus space left parenthesis B with rightwards harpoon with barb upwards on top plus C with rightwards harpoon with barb upwards on top right parenthesis

Therefore, the vector addition is assosciative. 

               

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