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** = (A

_{x}, TO

_{y}), knowing its coordinates, the following formula is used:

|** A**| = √ [(A

_{x})

^{2 }+ (A

_{y})

^{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} = (A_{x})^{2 }+ (A_{y})^{2}

That is to say,

|** A**| = √ [(A

_{x})

^{2 }+ (A

_{y})

^{2}]

In three dimensions

|** A**| = √ [(A

_{x})

^{2 }+ (A

_{y})

^{2}+ (A

_{z})

^{2 }]

**Exercises**

- 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**| = √ [(A

_{x})

^{2 }+ (A

_{y})

^{2 }+ (A

_{z})

^{2 }] = √ [(2)

^{2 }+ (4)

^{2 }+(5)

^{2 }]

= √ [4 + 16 + 25] =

= 6,71

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

In this case we have two points (A_{x1}, TO_{y1}) already_{x2}, TO_{y2}) that correspond to the ends of vector A; making use of the formula to determine magnitude of a vector, we have:

|** A**| = √ [(A

_{x2 }- TO

_{x1})

^{2 }+ (A

_{y2 }- TO

_{y1})

^{2}] = √ [(1 - 3)

^{2 }+ [2 – (-2)]

^{2 }] =

= √ [(-2)^{2} + (4)^{2 }] =

= √ 20 =

= 4,47

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

In this case we have two points (A_{x1}, TO_{y1}, TO_{z1}) already_{x2}, TO_{y2}, TO_{z2}) that correspond to the extremes of the vector ** A**; making use of the formula to determine magnitude of a vector, we have:

|** A**| = √ [(A

_{x2 }- TO

_{x1})

^{2 }+ (A

_{y2 }- TO

_{y1})

^{2 }+ (A

_{z2 }- TO

_{z1})

^{2}] = √ [(5 - 1)

^{2 }+ (7 - 3)

^{2 }+ (6 - 2)

^{2}] =

= √ [(4)^{2} + (4)^{2} + (4)^{2 }^{ }] =

= √ 48 =

= 6,93

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

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**| = √ [(A

_{x})

^{2 }+ (A

_{y})

^{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**| = √ [(A

_{x})

^{2 }+ (A

_{y})

^{2 }+ (A

_{z})

^{2 }] = √ [(6)

^{2 }+ (2)

^{2 }+(-3)

^{2 }]

= √ (36 + 4 + 9) =

= √49

= 7