What is Vector, coordinates?

A vector $\vec u$ represents a displacement: all arrows with the same direction, orientation and length represent the same vector. In a coordinate system, $\vec u\,(x, y)$ records the displacement ($x$ horizontally, $y$ vertically); for points $A(x_A, y_A)$ and $B(x_B, y_B)$,

Vectors add coordinate-wise, and $\lambda\vec u$ has coordinates $(\lambda x, \lambda y)$.