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How does coordinate-by-coordinate addition produce the sum of two vectors?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In linear algebra, vectors are often represented using coordinates that describe their position relative to a reference frame. When vectors are expressed in this component form, coordinate-by-coordinate addition provides a simple algebraic method for calculating their sum.1

If you have two vectors in two dimensions, represented as $(a_{1}, a_{2})$ and $(b_{1}, b_{2})$ , their sum is defined by adding their corresponding coordinates:
$(a_{1} + b_{1}, a_{2} + b_{2})$ .1

This concept extends to any number of dimensions. For example, in three dimensions, if two vectors are represented as $(x_{1}, y_{1}, z_{1})$ and $(x_{2}, y_{2}, z_{2})$ , their sum is the vector determined by $(x_{1} + x_{2}, y_{1} + y_{2}, z_{1} + z_{2})$ .21

This algebraic approach is equivalent to the geometric representation of vector addition, where the sum corresponds to the diagonal of a parallelogram formed by the two vectors as adjacent sides. Using coordinates allows for efficient computation in higher-dimensional spaces where geometric visualization is no longer practical.21