# Vectors in the Plane

A vector is a line segment running from point A (tail) to point B (head).

Each vector has a magnitude (also referred to as length) and a direction.

### Direction of a Vector

This is the direction of the line which contains the vector or any line which is parallel to it.

### Magnitude of a Vector

The magnitude of the vector is the length of the line segment . It is denoted by .

The magnitude of a vector is always a positive number or zero.

The magnitude of a vector can be calculated if the coordinates of the endpoints are known:

#### Examples

Calculate the magnitude of the following vectors:

Calculate the value of k knowing the magnitude of the vector = (k, 3) is 5.

### Position Vector

The vector that joins the coordinates origin, **O**, with a point, **P**, is the position vector of the point P.

### Components or Coordinates of a Vector

If the coordinates of A and B are:

#### Examples

Find the components of the vector :

The vector has the components (5, −2). Find the coordinates of A if the terminal point is known as B(12, −3).

Calculate the coordinates of Point D so that the quadrilateral of Points: A(−1, −2), B(4, −1), C(5, 2) and D form a parallelogram.