# Magnitude of a vector

A vector is a line segment oriented by an arrowhead drawn at one of its ends:

Point A is called origin and the point of the arrow (B) is called extreme vector.

A vector represents the magnitude and orientation of a physical quantity, therefore it has a length (a non-negative real number) as well as a direction (or orientation). The length of the vector is called magnitude or module.

## Definition of magnitude of a vector

The magnitude or module of a vector is the distance between the start point and the end point. In symbols the magnitude of the vector AB it is defined as | AB |.

The magnitude or module is the length proportional to the value of the vector.

## Calculating the magnitude of a vector

To calculate the magnitude or modulus of a vector A = (Ax, TOy), knowing its coordinates, the following formula is used:

|A| = √ [(Ax)2 + (Ay)2 ]

This expression is an application of the Pythagorean Theorem.

Let 0AxA be a right triangle, let us observe that 0A is the hypotenuse; applying the Pythagorean theorem we have:

(0A)2 = (Ax)2 + (Ay)2

That is to say,

|A| = √ [(Ax)2 + (Ay)2 ]

In three dimensions

|A| = √ [(Ax)2 + (Ay)2 + (Az)2 ]

## Exercises

1. Calculate the magnitude of the vector A whose position is given by (2, 4, 5).

We see that 2 is the component "x", 4 is "y" and 5 is "z". Now, using the formula to determine the magnitude of a vector, we have:

|A| = √ [(Ax)2 + (Ay)2 + (Az)2 ] = √ [(2)2 + (4)2 +(5)2 ]

= √ [4 + 16 + 25] =

= 6,71

1. Calculate the magnitude of the vector A whose ends are (3, -2) and (1, 2).

In this case we have two points (Ax1, TOy1) alreadyx2, TOy2) that correspond to the ends of vector A; making use of the formula to determine magnitude of a vector, we have:

|A| = √ [(Ax2 - TOx1 )2 + (Ay2 - TOy1)2] = √ [(1 - 3)2 + [2 – (-2)]2 ] =

= √ [(-2)2 + (4)2 ] =

= √ 20 =

= 4,47

1. Calculate the magnitude of the vector A whose ends are (1, 3, 2) and (5, 7, 6).

In this case we have two points (Ax1, TOy1, TOz1) alreadyx2, TOy2, TOz2) that correspond to the extremes of the vector  A; making use of the formula to determine magnitude of a vector, we have:

|A| = √ [(Ax2 - TOx1 )2 + (Ay2 - TOy1)2 + (Az2 - TOz1)2] = √ [(5 - 1)2 + (7 - 3)2 + (6 - 2)2] =

= √ [(4)2 + (4)2 + (4)2  ] =

= √ 48 =

= 6,93

1. Calculate the magnitude of the vectors A = (9, 7) and A = (6, 2, -3).

We see that for the vector A, 9 is the component "x" and 7 is "y". Now, using the formula to determine the magnitude of a vector, we have:

|A| = √ [(Ax)2 + (Ay)2] = √ [(9)2 + (7)2 ]

= √ (81 + 49) =

= 11,4

Now, we see that for the vector B, 6 is the component "x", 2 "y" and -3 is "z". Now, using the formula to determine the magnitude of a vector, we have:

|A| = √ [(Ax)2 + (Ay)2 + (Az)2 ] = √ [(6)2 + (2)2 +(-3)2 ]

= √ (36 + 4 + 9) =

= √49

= 7