Vector

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A vector of dimension n is a sequence or finite ordered list of n components, these being real numbers, which is represented by a line segment and is used to format physical quantities.

In other words, a vector of dimension n is a row that contains n real numbers, it is represented through a segment with sense and direction and, it serves to represent physical quantities such as volume, pressure, energy ...

Formula of a vector

Given a vector u of dimension n in the space Rn it is represented as

Vector of dimension n that belongs to the space Rn

As described, the vector is a row where its components are real numbers. This row is finite since the vector has an ending and is the component with the subscript n.

Vectors and arrows

Vectors and arrows have a lot in common! Can you tell which of the following pictograms is a vector?

Arrows

The two pictograms can be a vector and an arrow at the same time because they both have a direction, a sense, and a magnitude. So, to easily remember what a vector is, let's think of an arrow.

Characteristics

To be a vector, it must have direction, sense, and magnitude.

  • Direction: Like arrows when we shoot them with a bow, vectors also have direction and it is the angle of the vector that is formed with respect to the horizontal axis. In other words, the direction is indicated by the slope that is formed if we draw a thin (straight) line above the arrow.
Address
  • Direction: It is the orientation of the segment and can be positive or negative. The direction is indicated by the pointed part of the arrow.
Sense
  • Magnitude: The magnitude is the size of the arrow, that is, of the vector.

Magnitude

Mathematical notation of vectors

In this case we use a vector called p and another vector called r. Vector p begins at point P and ends at point Q. Vector r begins at point R and ends at point S.

Vectors p and q

In the same line of the comparison with an arrow, a vector is expressed by means of the extreme points and drawing a small arrow above these points. So, this arrow indicates the starting point from which the vector begins to where it ends.

Representation of a vector

Vector with coordinates in the xy plane

In this case, the vector u is represented in the Cartesian plane and is indicated through the coordinates uy and ux.

Mathematical expression of the vector from the coordinates

Scalar and vector

The main difference between a scalar and a vector is that a scalar has neither direction nor meaning. In other words, a scalar will only have magnitude.

App

The vectors are found in the daily life of mathematics and in all the sciences that depend on them, whether they are statistics, physics, engineering ...

Example

Draw a vector p in the Cartesian plane given the coordinates.

Example of a vector

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