A vector is a quantity that has a length (a non-negative real number), as well as direction and a sense. A vector represents the magnitude and orientation of a physical quantity.

Two vectors ** A** y

**they are equal when they have the same magnitude, the same direction and the same sense; are**

*B***negatives**(or

**opposites**) when they have the same magnitude (modulus) and direction, but

**Wrong Way**. We can perform addition, subtraction, multiplication, and division operations with these negative vectors.

**Definition**

Sean ** A** y

**two vectors; we will say that**

*B***is the vector negative or opposite to**

*B***if it has the same magnitude and direction, but opposite direction:**

*A*** A **= –

*B*Let's see if we compare ** B** with

**They both have the same magnitude (or modulus) and direction, but in the opposite direction.**

*A**It is called the negative (or opposite) vector of A, to the vector B which has the same magnitude (m*

*or*

*dulo) and the same address*

*or*

*n, but opposite sense.*

**Exercises**

- Determine the negative (or opposite) vector of the following vectors:
= (3, -5) and*A*= (1, 6).*B*

Be ** a** the negative vector a

**, Thus:**

*A*** a **= –

*A*That is to say:

** a **= – (3, -5) = (-3, 5)

Similarly, be ** b** the negative vector a

**, Thus:**

*B*** b **= –

*B*That is to say:

** b **= – (1, 6) = (-1, -6)

- Since the end points of two vectors are A (2, 3), B (4, -7), C (10, -8), D (8,2). Show that the vectors
y*AB*they are negative (or opposite) to each other.*CD*

We know that ** AB** y

**are negative (or opposite) vectors of each other, if:**

*CD*- They have the same magnitude (modulus).
- They have the same address.
- They have the opposite sense.

__Calculation of the modules of both vectors: __

We have two points A (2, 3) and B (4, -7) that correspond to the ends of the vector ** AB**; making use of the formula to determine magnitude of a vector, we have:

|*A*** B**| = √ [(B

_{x }- TO

_{x})

^{2 }+ (B

_{y }- TO

_{y})

^{2 }] = √ [(4 - 2)

^{2 }+ (-7 - 3)

^{2 }] =

= √ [(2)^{2} + (-10)^{2}^{ }] =

= √ (4 + 100) =

= √104

Now, we also have two points C (10, -8) and D (8,2) that correspond to the extremes of the vector ** CD**; we have:

|** CD**| = √ [(8 - 10)

^{2 }+ (2 + 8)

^{2 }] =

= √ [(-2)^{2} + (10)^{2}^{ }] =

= √ (4 + 100) =

= √104

In consecuense:

|*A*** B**| = |

**|**

*CD*Now, we calculate the slopes of the vectors; if you are the same, then, it is true that ** AB** y

**it has the same address:**

*CD*m**_{AB}** = [(B

_{y }- TO

_{y}) / (B

_{x }- TO

_{x})] = [(-7 - 3) / (4 - 2)] = - 10/2 = - 5

m**_{CD}** = [(D

_{y }- C

_{y}) / (D

_{x }- C

_{x}) ] = [ (2 + 8) / (8

_{ }- 10)] = 10 / - 2 = - 5

To check that they make opposite sense, we analyze their x and y components:

** AB **= (B

_{x }- TO

_{x}, B

_{y }- TO

_{y}) = (2, -10)

** CD** = (D

_{x }- C

_{x}, D

_{y }- C

_{y}) = (-2,10)

From here we see that:

** AB **= –

*CD*Therefore, both vectors are negative vectors with respect to each other.