The magnitude of a vector
is the distance between the initial point *P * and the end point *Q *. In symbols the magnitude of is written as .

If the coordinates of the initial point and the end point of a vector is given, the Distance Formula can be used to find its magnitude.

**Example :**

Find the magnitude of the vector whose initial point *P * is at (1, 1) and end point is at *Q * is at (5, 3).

Solution:

Use the Distance Formula.

Substitute the values of *x*_{1} , *y*_{1} , *x*_{2} , and *y*_{2} .

The magnitude of is about 4.5.

The direction of a vector is the measure of the angle it makes with a horizontal line.

One of the following formulas can be used to find the direction of a vector:

, where *x * is the horizontal change and *y * is the vertical change

or

,
where ( *x*_{1} , *y*_{1} ) is the initial point and ( *x*_{2} , *y*_{2} ) is the terminal point.

**Example :**

Find the direction of the vector whose initial point *P * is at (2, 3) and end point is at *Q * is at (5, 8).

The coordinates of the initial point and the terminal point are given. Substitute them in the formula .

Find the inverse tan, then use a calculator.

The vector has a direction of about 59° .