A vector represents a quantity with direction and magnitude. Geometrically, it can be pictured as an arrow: moving from one point to another produces a displacement vector.

In coordinates, a vector is written by its components. In two dimensions,

$$\mathbf v=(v_x,v_y),$$

and in three dimensions,

$$\mathbf v=(v_x,v_y,v_z).$$

The components record how much the vector points along each coordinate axis.

Points and vectors

The same ordered-pair notation is used for points and vectors, but the interpretation differs. A point $(3,2)$ describes a position, while a vector

$$\mathbf v=(3,2)$$

describes a displacement of three units in the $x$ direction and two in the $y$ direction.

A vector does not depend on where its arrow is drawn: arrows with the same components represent the same vector.

Dimension and components

A vector in $\mathbb R^n$ has $n$ real components:

$$\mathbf v=(v_1,v_2,\ldots,v_n).$$

Two vectors are equal when all corresponding components are equal.

The zero vector

$$\mathbf0=(0,0,\ldots,0)$$

represents no displacement. These component descriptions let geometric ideas be handled algebraically.

Visual intuition: vectors