What do collinear forces mean

Colinear system and examples

The colinear vectors They are one of three types of vectors in existence. It is about those vectors that are in the same direction or line of action. This means the following: Two or more vectors are collinear if they are arranged in straight lines that are parallel to one another.

A vector is defined as a quantity applied to a body and is characterized by having a direction, a meaning and a scale. The vectors can be found in the plane or in space and can be of several types: collinear vectors, competing vectors, and parallel vectors.

index

  • 1 colineal vectors
  • 2 properties
    • 2.1 Example 1
    • 2.2 Example 2
    • 2.3 Example 1
  • 3 collinear vector system
    • 3.1 Collinear vectors with opposite senses
    • 3.2 Collinear vectors with the same sense
    • 3.3 Collinear vectors with equal sizes and opposite senses
  • 4 Difference Between Colinear and Competing Vectors
  • 5 references

Collinear vectors

The vectors are collinear when the line of action of one is exactly the same line of action of all other vectors, regardless of the size and direction of each of the vectors.

Vectors are used as representations in various fields like math, physics, algebra and also in geometry where the vectors are only collinear if their direction is the same regardless of their meaning.

properties

- Two or more vectors are collinear if the relationship between the coordinates is the same.

example 1

We have the vectors m = {m_x; m_y} and n = {n_x; n_y}. These are collinear if:

Example 2

It can be determined whether the vectors j = {3,6,15} and p = {1,2,5} are collinear by the relationship of their coordinates, which must be proportional to one another; This is:

- Two or more vectors are collinear when the product or vector multiplication is zero (0). This is because, in the coordinate system, each vector is identified by its respective coordinates, and when these are proportional to each other, the vectors are collinear. This is expressed as follows:

example 1

We have the vectors a = (10, 5) and b = (6, 3). To determine whether they are collinear, determinant theory is applied, which establishes the equality of the cross products. This way you have to:

Colinear vector system

Collinear vectors are displayed graphically, showing the direction and meaning of the same - taking into account that they must pass through the point of application - and the module, which has a certain scale or length.

The system of collinear vectors is formed when two or more vectors act on an object or a body, represent a force and act in the same direction.

For example, when two collinear forces are applied to a body, the resultant depends only on the sense in which they act. There are three cases which are:

Collinear vectors with opposite senses

The resultant of two collinear vectors is equal to the sum of these:

R = Σ F = F1 + F2.

example

When two forces act on a carriage F.1 = 40 N and F2 = 20 N in the opposite direction (as shown in the picture), the result is:

R = Σ F = (- 40 N) + 20 N.

R = - 20 N.

The negative sign expresses that the body moves to the left with a force of 20 N.

Collinear vectors with the same sense

The magnitude of the resulting force is equal to the sum of the collinear vectors:

R = Σ F = F1 + F2.

example

When two forces act on a carriage F.1 = 35 N and F2 = 55 N in the same direction (as shown in the picture) the result is:

R = Σ F = 35 N + 55 N.

R = 90 N.

The positive result indicates that the collinear vectors act to the left.

Collinear vectors with equal sizes and opposite senses

The resultant of the two collinear vectors is equal to the sum of the collinear vectors:

R = Σ F = F1 + F2.

Since the forces are the same size but in the opposite direction - that is, one becomes positive and the other negative - when the two forces are added the result will be zero.

example

When two forces act on a carriage F.1 = -7 N and F2 = 7 N that are the same size but in the opposite direction (as shown in the picture) the result is:

R = Σ F = (-7 N) + 7N.

R = 0

Since the resultant equals 0, this means that the vectors balance each other and the body is therefore in equilibrium or at rest (it will not move).

Difference Between Colinear and Competing Vectors

Collinear vectors are characterized by having the same direction in the same line or because they are parallel to a line; That is, they are vectors that direct parallel lines.

On the other hand, the competing vectors are defined because they are in different lines of action that are intercepted in a single point.

In other words, regardless of module, direction or direction, they have the same reference or arrival location and form an angle between them.

The systems of competing vectors are solved by mathematical methods or graphs, which are the method of the parallelogram of forces and the method of the polygon of forces. This determines the value of a resulting vector that indicates the direction in which a body moves.

Basically, the main difference between the colinear vectors and the competing vectors is the line of action in which they act: the collinear vectors act in the same line, while the competing vectors occur in different lines.

That is, the collinear vectors act in a single plane, "X" or "Y"; and the simultaneous act in both planes, starting from the same point.

Collinear vectors are not found in a point, as are the parallel vectors because they are parallel to each other.

In the left picture you can see a block. It is tied with a rope and the knot divides it into two parts; When pulled in different orientations and with different forces, the block will move in the same direction.

Two vectors that match in one point (the block) are represented regardless of module, sense or direction.

Instead, a roller appears in the right picture that lifts a box. The rope represents the line of action; When it is pulled, two forces (vectors) act on it: one force of tension (when climbing the block) and another force that exerts the weight of the block. Both have the same direction, but in opposite directions; they do not agree on one point.

credentials

  1. Estalella, J.J. (1988). Vector analysis. part 1
  2. Gupta, A. (s.f.). Tata McGraw-Hill training.
  3. Jin Ho Kwak, S. H. (2015). Linear Algebra. Springer science and business media.
  4. Montiel, H.P. (2000). Physics 1 for technological high school diploma. Grupo editorial team Patria.
  5. Santiago Burbano de Ercilla, C. G. (2003). General Physics Editorial Tebar.
  6. Sinha, K. (s.f.). A Textbook of Mathematics XII Vol. 2. Rastogi Publications.