7 junio, 2024

Elements of a vector: what they are and their characteristics

The elements of a vector They are direction, distance, and modulus. In mathematics, physics, and engineering, a vector is a geometric object that has a magnitude (or length) and a direction. According to vector algebra, vectors can be added to other vectors.

Frequently, a vector is represented by a line segment with a definite direction, or graphically represented as an arrow, connecting an initial point A with a terminal point B, denoted by AB.

A vector is what is needed to get point A to point B. Vectors play an important role in physics: the speed and acceleration of a moving object and the forces acting on it can be described with vectors.

Many other physical qualities can be thought of as vectors. The mathematical representation of a physical vector depends on the coordinate system used to describe it.

There are several classes of vectors, among them we can find sliding vectors, collinear vectors, concurrent vectors, position vectors, free vectors, parallel vectors, and coplanar vectors, among others.

elements of a vector

Mainly, a vector has three elements: the direction, the sense and the module.

Address

A vector is an entity that has magnitude and direction. Examples of vectors include displacement, velocity, acceleration, and force. To describe one of these vector quantities, it is necessary to find the magnitude and the direction.

For example, if the speed of an object is 25 meters per second, then the description of the object’s speed is incomplete, since the object may be moving at 25 meters per second south, or 25 meters per second north, or 25 meters per second to the southeast.

In order to fully describe the speed of an object, both must be defined: both the magnitude of 25 meters per second, and the direction, such as south.

For such descriptions of vector quantities to be useful, it is important that everyone agrees on how the direction of the object is described.

Most people are used to the idea that on a map east is referred to by looking to the right. But this is just a convention that map makers have used for years so everyone can agree.

So what is the direction of a vector quantity that is not heading north or east but somewhere between north and east? For these cases it is important that there is some convention to describe the direction of said vector.

This convention is referred to as the CCW. Using this convention, we can describe the direction of any vector in terms of its angle of rotation to the left.

Using this convention, the north direction would be 90°, since if a vector is pointing east it would have to be rotated 90° to the left to reach the north point.

Likewise, the west direction would be located at 180°, since a west-pointing vector would have to be rotated 180° to the left to point to the west point.

In other words, the direction of a vector will be represented through a line contained in the vector or any line that is parallel to it.

It will be determined by the angle formed between the vector and any other reference line. That is, the direction of the line that is in the vector or some line parallel to it is the direction of the vector.

Sense

The direction of the vector refers to the element that describes how point A goes towards end B:

The sense of a vector is specified by the order of two points on a line parallel to the vector, unlike the direction of the vector, which is specified by the relationship between the vector and any reference lines and/or planes.

Both orientation and sense determine the direction of a vector. Orientation tells what angle the vector is at, and sense tells where it is pointing.

The direction of the vector only establishes the angle that a vector makes with its horizontal axis, but that can create ambiguity, since the arrow can point in two opposite directions and still make the same angle.

The sense clarifies this ambiguity and indicates where the arrow is pointing or where the vector is going.

Somehow, the sense tells us the order in which to read the vector. Indicates where the vector begins and ends.

Module

The module, or amplitude of a vector, can be defined as the length of the segment AB. The module can be represented through a length that is proportional to the value that the vector has. The magnitude of a vector will always be zero, or in other cases some positive number.

In mathematics, the vector will be defined by its Euclidean distance (module), direction and direction.

The Euclidean distance, or Euclidean distance, is the ‘ordinary’ distance in a straight line between two points located in a Euclidean space. With this distance, Euclidean space becomes metric space.

A Euclidean distance between two points, say P and Q, is the distance between the line segment connecting them:

The position of a point in a Euclidean space n is a vector. Thus P and Q are vectors, starting from the origin of space and their points indicating two points.

The Euclidean norm, magnitude, or Euclidean distance of a vector measures the length of that vector.

References

Vector direction. Retrieved from physicsclassroom.com. What is the sense of a vector? Retrieved from physics.stackexchange.com.

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