We explain what collinear vectors are, the system of collinear vectors and we give several examples

**What are collinear vectors?**

The **collinear vectors** They are one of the three types of vectors that exist. These are those vectors that are in the same direction or line of action. This means the following: two or more vectors will be collinear if it happens that they are arranged in lines that are parallel to each other.

A vector is defined as a magnitude applied to a body and is characterized by having a direction, a sense and a scale. Vectors can be found in the plane or in space and can be of different types: collinear vectors, concurrent vectors, and parallel vectors.

**When are there collinear vectors?**

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

Vectors are used as representations in different areas such as mathematics, physics, algebra, and also in geometry, where vectors are collinear only when their direction is the same, regardless of whether their direction is not.

**Examples of collinear vectors**

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

**Example 1**

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

**Example 2**

It can be determined if the vectors j={3,6,15} and p={1,2,5} are collinear through the relationship of their coordinates, which must be proportional to each other; that is to say:

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

**Example 1**

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

**System of collinear vectors**

The collinear vectors are graphically represented using their direction and direction —bearing in mind that they must pass through the point of application— and the module, which is a scale or determined length.

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

For example, if two collinear forces are applied to a body, the result of these will only depend on the direction in which they act. There are three cases, which are:

**Collinear vectors with opposite directions**

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

R = ∑ F = F1 + F2.

**Example**

If two forces F1 = 40 N and F2 = 20 N act on a cart in the opposite direction (as shown in the image), the resultant is:

R = ∑ F = (- 40N) + 20N.

R = – 20N.

The negative sign expresses that the body will move to the left, with a force equivalent to 20 N.

**Collinear vectors with the same sense**

The magnitude of the resultant force will be equal to the sum of the collinear vectors:

R = ∑ F = F1 + F2.

**Example**

If two forces F1 = 35 N and F2 = 55 N act on a cart in the same direction (as shown in the image), the resultant is:

R = ∑ F = 35N + 55N.

R = 90N.

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

**Collinear vectors with equal magnitudes and opposite directions**

The resultant of the two collinear vectors will be equal to the sum of the collinear vectors:

R = ∑ F = F1 + F2.

Since the forces have the same magnitude but in the opposite direction —that is, one will be positive and the other negative—, when the two forces are added, the resultant will be equal to zero.

**Example**

If two forces act on a cart F1 = -7 N and F2 = 7 N, which have the same magnitude, but in the opposite direction (as shown in the image), the resultant is:

R = ∑ F = (-7N) + 7N.

R = 0.

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

**Difference Between Collinear and Concurrent 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 direction vectors of parallel lines.

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

In other words, they have the same point of origin or arrival —regardless of their module, sense or direction—, forming an angle between them.

Concurrent vector systems are solved by mathematical or graphical methods, which are the force parallelogram method and the force polygon method. Through these, the value of a resulting vector will be determined, which indicates the direction in which a body will move.

Basically, the main difference between collinear and concurrent vectors is the line of action in which they act: collinear ones act in the same line, while concurrent ones in different ones.

That is, the collinear vectors act in a single plane, “X” or “Y”; and the concurrents act on both planes, starting from the same point.

Collinear vectors do not meet at a point, as concurrent vectors do, because they are parallel to each other.

In the left image you can see a block. It is tied with a rope and the knot divides it in two; when pulled in different directions and with different forces, the block will move in the same direction.

Two vectors are being represented that meet at a point (the block), regardless of their module, sense or direction.

On the other hand, in the right image there is a pulley that lifts a box. The string represents the line of action; when it is pulled, two forces (vectors) act on it: a tension force (as the block rises) and another force, the force exerted by the weight of the block. Both have the same direction, but in opposite directions; They do not meet at one point.