Magnitude and Direction of Vectors

Magnitude of a Vector

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

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.

| PQ |= ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2

Example :

Find the magnitude of the vector PQ 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 .

| PQ |= ( 51 ) 2 + ( 31 ) 2 = 4 2 + 2 2 = 16+4 = 20 4.5

The magnitude of PQ 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:

tanθ= y x , where x is the horizontal change and y is the vertical change

or

tanθ= y 2 y 1 x 2 x 1 , 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 PQ 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 tanθ= y 2 y 1 x 2 x 1 .

tanθ= 83 52 = 5 3

Find the inverse tan, then use a calculator.

θ= tan 1 ( 5 3 ) 59°

The vector PQ has a direction of about 59° .