Why are the vector coordinates calculated this way?

Question

Suppose we are given a vector $\vec{AB}$, where $A(x_1;y_1)$ and $B(x_2;y_2)$. Why do we calculate its coordinates as $\vec{AB}(x_2-x_1;y_2-y_1)$? And how can a segment have "coordinates" at all?

Answer 1

As suggested in the comments, given two points $A=(x_1;y_1)$ and $B=(x_2;y_2)$, by definition vector $\vec{AB}$ is the vector such that

$$\vec{OA}+\vec{AB}=\vec{OB}$$

therefore

$$\vec{AB}=\vec{OB}-\vec{OA}=(x_2-x_1;y_2-y_1)$$

that is vector $\vec{AB}$ has components $x_2-x_1$ along $x$ axis and $y_2-y_1$ along $y$ axis.

Note that $\vec{AB}$ is not a segment but we can consider it as a directed line segment in the plane or more in general in the space.

See also the related

• Vectors and directed line segments