Vector Examples - encyclopedia

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Elements of a vector
Types of vectors
Vectors in two and three dimensions
Graphic representation of vectors
Examples of vector quantities in physics
Examples of vectors in mathematics

A vector is a mathematical tool, generally used in geometry and physics, that allows to carry out calculations and operations.

In physics, a vector is a line segment in space that has modulus (also called length) and direction (or orientation). Vectors are represented graphically with an arrow and help describe vector quantities.

Vector magnitudes are represented through a vector because they cannot be determined by a single real number but it is necessary to know its direction and sense. For example: speed, displacement. This distinguishes them from scalar quantities, which only require a number and a certain unit of measure to be defined, for example: lpressure, volume, temperature.

Continue on: Vector and scalar quantities

In mathematics, vectors are the elements of a vector space. This notion is more abstract, since in many vector spaces the vector cannot be defined from module and direction, for example: vectors in infinite-dimensional spaces. The representation used to represent a vector in a space of "n" dimensions is:v= (a₁,to₂,to₃,…to_n)

Vectors can be added or subtracted from each other, giving rise to a new resulting vector, or multiplied by a scalar, vector or mixed value.

Elements of a vector

To define a vector completely, three characteristics must be specified that distinguish one vector from another:

Module. It is determined by the length or length of the line segment.
Address. It is determined by the orientation of the line in the plane.
Sense. It is determined by the origin and endpoint of the line segment.

Types of vectors

Different classes of vectors can be distinguished according to the characteristics they present and the relationship they have with other vectors:

Unit vectors. Vectors whose modulus is equal to 1.
Free vectors. Vectors that are not applied at any particular point.
Sliding vectors. Vectors whose point of application slides along the line of action.
Fixed vectors (or linked vectors). Vectors that are applied to a particular point.
Collinear vectors. Two or more vectors that act on the same action line.
Concurrent vectors (or angular vectors). Two or more vectors whose directions pass through the same point, forming an angle when the rays intersect.
Parallel vectors. Two or more vectors acting on a rigid body with parallel lines of action.
Opposite vectors. Vectors that have the same direction and the same module, but that have opposite directions.
Coplanar vectors. Vectors whose lines of action lie in the same plane.
Resulting vectors.Given a system of vectors, it is the vector that produces the same effect as all the component vectors of the system.
Balancing vectors.Vector with the same magnitude and direction as the resulting vector, but with the opposite sense.

Vectors in two and three dimensions

Vectors can be represented in two-dimensional ("x", "y") or three-dimensional ("x", "y", "z") spaces. In any case, the vectors can be defined by their coordinates on each of the axes.

In the case of a two-dimensional space, any vector can be defined as: v = (v_x, v_Y). The terms in parentheses are the coordinates on the "x" and "y" axes.

On the other hand, in a three-dimensional space, a vector is defined as: v = (v_x, v_Y, v_z). One more coordinate is added to indicate the coordinate on the "z" axis.

Graphic representation of vectors

Vectors are represented in a general way using a two- or three-dimensional plane.

First, the support or direction line is graphed, on which several vectors can exist, drawing a line segment that arises from the origin.
Second, the length of the vector is marked, which is determined by the module (the greater the module, the longer the ray), and which is directed to a direction or point of application (which is why the vectors are drawn as arrows pointing in the direction in question).
Finally, the name of the vector is written on the application point.