# How is the resultant displacement affected when two displacement vectors are added in a different order?

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember that the resulting vector `barr`  is not influenced by the order of addition of two vectors `bar (v_1)`  and `bar(v_2).`

`bar (v_1) +bar(v_2) = bar (v_2)+ bar(v_1) = bar r`

You may use analytical method to prove the independence of resulting vector of order of addition such that:

`bar (v_1) = v_(1x) bar i + v_(1y) bar j`

`bar (v_2) = v_(2x) bar i + v_(2y) bar j`

`bar r = r_x bar i + r_y bar j`

`bar (v_1) +bar(v_2) = v_(1x) bar i + v_(1y) bar j+v_(2x) bar i + v_(2y) bar j`

`bar (v_1) +bar(v_2) = (v_(1x)+v_(2x)) bar i + (v_(1y)+v_(2y)) bar j`

Substituting `r_x`  for `(v_(1x)+v_(2x))`  and `r_y`  for`(v_(1y)+v_(2y))`  yields:

`bar (v_1) +bar(v_2) =r_x bar i +r_y bar j`

Hence, by commutative law of addition, the resultant displacement vector is the same, regardless the order of addition of the vectors.