Magnitude and Direction of Vectors

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

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.

| P Q → | = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2

Example 1:

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

Use the Distance Formula.

Substitute the values of x 1 , y 1 , x 2 , and y 2 .

| P Q → | = ( 5 − 1 ) 2 + ( 3 − 1 ) 2 = 4 2 + 2 2 = 16 + 4 = 20 ≈ 4.5

The magnitude of P Q → 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:

tan θ = y x , where x is the horizontal change and y is the vertical change

tan θ = 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 2:

Find the direction of the vector P Q → 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 tan θ = y 2 − y 1 x 2 − x 1 .

tan θ = 8 − 3 5 − 2 = 5 3

Find the inverse tan, then use a calculator.

θ = tan − 1 ( 5 3 ) ≈ 59 °

The vector P Q → has a direction of about 59 ° .

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