How would I use a diagram to prove the following properties of vector addition?

How would I use a diagram to prove the commutative and associative properties of vector addition?


A) Vector addition is commutative: a + b = b + a


B) Vector addition is associative: a + (b + c) = (a + b) + c





How would I draw each diagram to prove this? A detailed explanation of a suitable diagram would be awesome!|||Easy. Start at the origin of the cartesian plane, and draw Vector A (which can be any vector you can imagine. Then, start at the end of Vector A and draw Vector B (which can also be any vector in any direction.)





Then, re-draw starting with Vector B at the origin, and Vector A beginning at the end of Vector A.





Another way to work on this is to show that Vector A and B's lengths can be broken up as (L cos 螛, L sin 螛) where L is the length of the vector and 螛 = the angle from the X-axis.|||A) Draw a parallelogram OACB with one corner at the origin O, and vectors OA = a, OB = b.





Since OACB is a parallelogram, OA = BC = a and OB = AC = b as vectors. So





a + b


= OA + AC


= OC


= OB + BC


= b + a.





B) Draw a quadrilateral OABC, with OA = a, AB = b, BC = c. Then





a + (b + c)


= OA + (AB + BC)


= OA + AC


= OC


= OB + BC


= (OA + AB) + BC


= (a +b) + c.