# Negative Vectors

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 B they are equal when they have the same magnitude, the same direction and the same sense; are 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 B two vectors; we will say that B is the vector negative or opposite to A if it has the same magnitude and direction, but opposite direction:

A = – B

Let's see if we compare B with AThey both have the same magnitude (or modulus) and direction, but in the opposite direction.

It is called the negative (or opposite) vector of A, to the vector B which has the same magnitude (mordulo) and the same addressorn, but opposite sense.

## Exercises

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

Be a the negative vector a A, Thus:

a = – A

That is to say:

a = – (3, -5) = (-3, 5)

Similarly, be b the negative vector a B, Thus:

b = – B

That is to say:

b = – (1, 6) = (-1, -6)

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

We know that AB y CD are negative (or opposite) vectors of each other, if:

• 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:

|AB| = √ [(Bx - TOx )2 + (By - TOy)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:

|AB| = |CD|

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

mAB = [(By - TOy) / (Bx - TOx)] = [(-7 - 3) / (4 - 2)] = - 10/2 = - 5

mCD = [(Dy - Cy) / (Dx - Cx) ] = [ (2 + 8) / (8  - 10)] = 10 / - 2 = - 5

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

AB = (Bx - TOx, By - TOy) = (2, -10)

CD = (Dx - Cx, Dy - Cy) = (-2,10)

From here we see that:

AB = – CD

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