A vector v is a quantity that has both magnitude and direction. We picture a vector as an arrow from an initial point O to a terminal point P with this provision: arrows that have the same length (magnitude) and direction represent the same vector (Fig. 1).

The vector v of Figure 1 is also denoted by . We use boldface letters such as v to denote vectors. But because it is difficult to write boldface by hand, we suggest that you use as a substitute for v when you want to denote a vector by a single letter.

The **magnitude** of the vector v =, denoted by |v|, || or || is the length of the line segment OP. Two vectors have the **same direction** if they are parallel and point in the same direction. Two vectors have **opposite directions** if they are parallel and point in opposite directions. The **zero vector**, denoted by 0 or , has magnitude 0 and arbitrary direction. Two vectors are **equal** if they have the same magnitude and direction. So a vector can be translated from one location to another as long as the magnitude and direction do not change.

Any vector in a rectangular coordinate system can be translated so that its initial point is the origin O. The vector such that = is said to be the **standard****vector** for (Fig. 2). Note that is the standard vector for infinitely many vectors—all vectors with the same magnitude and direction as .

Given the coordinates of the endpoints of vector how do we find its corresponding standard vector ? The coordinates of the origin O, the initial point of OP, are always (0, 0). The coordinates of P, the terminal point of are given by

(x_{p}, y_{p}) = (x_{b} - x_{a}, y_{b} - y_{a})

where A = (x_{a}, y_{a}) and B = (x_{b}, y_{b}).

**EXAMPLE 1 Finding a Standard Vector for a Given Vector**

Given the geometric vector with initial point A = (3, 4) and terminal point B = (7, 1), find the coordinates of the point P such that =

**SOLUTION** The coordinates of P are given by

(x_{p}, y_{p}) = (x_{b} - x_{a}, y_{b} - y_{a})

= (7 - 3, -1 - 4)

= (4, -5)

Note in Figure 3 that if we start at A, then move to the right four units and down five units, we will be at B. If we start at the origin, then move to the right four units and down five units, we will be at P.

Example 1 suggests that there is a one-to-one correspondence between vectors in a rectangular coordinate system and points in the system. Any vector is completely specified by the point P = (x_{p}, y_{p}) such that = (we are not concerned that has a different position than ; we are free to translate a vector anywhere we please). Conversely, any point P of the system corresponds to the vector .

A vector can be denoted by an ordered pair of real numbers. To avoid confusion, we use {c, d} to denote the vector with initial point (0, 0) and terminal point (c, d) (Fig. 4). The real numbers c and d are called the **scalar components** of the vector {c, d}. Two vectors u = {a, b} and v = {c, d} are equal if their corresponding components are equal, that is, if a = c and b = d. The zero vector is 0 {0, 0}. The magnitude of the vector {a, b} is the length of the line segment from (0, 0) to (a, b) [Fig. 5].