Magnitude and Direction of Vectors

Magnitude of a Vector

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 x1 , y1 , x2 , and y2 .

The magnitude of is about 4.5.

Direction of a Vector

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 ( x1 , y1 ) is the initial point and ( x2 , y2 ) 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° .