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.

opposite vectorsTwo 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.

negative vectors - example