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Adding two vectors preserves the relation because the sum of the third components equals the sum of the first components plus the sum of the second components. Specifically, if $x_3 = x_1 + x_2$ and $y_3 = y_1 + y_2$, then $(x+y)_3 = x_3 + y_3 = (x_1 + x_2) + (y_1 + y_2) = (x_1 + y_1) + (x_2 + y_2) = (x+y)_1 + (x+y)_2$.

Conditions: $x$ and $y$ are vectors in $\mathbb{R}^3$.; $x$ and $y$ satisfy the equation $x_3 = x_1 + x_2$ and $y_3 = y_1 + y_2$ respectively.; Vector addition is defined componentwise.

Adding two vectors preserves the relation because the sum of the third components equals the sum of the first components plus the sum of the second components. Specifically, if $x_3 = x_1 + x_2$ and $y_3 = y_1 + y_2$, then $(x+y)_3 = x_3 + y_3 = (x_1 + x_2) + (y_1 + y_2) = (x_1 + y_1) + (x_2 + y_2) = (x+y)_1 + (x+y)_2$.

Conditions: $x$ and $y$ are vectors in $\mathbb{R}^3$.; $x$ and $y$ satisfy the equation $x_3 = x_1 + x_2$ and $y_3 = y_1 + y_2$ respectively.; Vector addition is defined componentwise.