▶ ▶ ▶ Vectors - Definition and Examples ✔✔✔

If we want to express the temperature in the city we will only need a number and a unit (“℃” or “℉”). But what if we want to express the location of an airplane in the air? Is it enough to say that it is "x" kilometers from the airport? No! We must specify the direction of its displacement (for example, the line that forms a certain angle with the horizontal) and its direction (North, South, East or West). Physical quantities that have magnitude, direction, and direction are called vector quantities and can be represented as a vector.

A vector quantity is represented by vectors, that is, by a magnitude and a direction (or orientation).

Vector definition

A vector is a line segment oriented by an arrowhead drawn at one of its ends:

vector

Point A is called origin and the point of the arrow (B) is called extreme vector.

What is a vector?

A vector is a quantity that has a length (a non-negative real number), as well as direction (or orientation). A vector represents the magnitude and orientation of a physical quantity.

Notation

A vector can be represented using various procedures:

  • They are generally represented in printed texts by letters in bold font, to differentiate them from the scalar magnitudes that are represented in italics.

vector   It is shown AB

  • Placing an arrow on top of the letters that determine the vector, for example:

vector   It is shown: \ LARGE \ vec {AB}

  • Using a single letter with an arrow above it:

vector It is shown:

\ LARGE \ vec {A}

\ LARGE \ vec {V} = \ left (V_ {1}, V_ {2}, ..., V_ {n} \ right) = \ left \ langle V_ {1}, V_ {2}, ... , V_ {n} \ right \ rangle

  • A three-dimensional vector can be represented as follows:

\ LARGE \ vec {V} = V_ {x} \ hat {i} + V_ {y} \ hat {j} + V_ {z} \ hat {k}

Where Vx, Vy y Vz are the components of a vector e "I J K" are the unit vectors (magnitude or modulus 1) in the direction "X and Z".

  • Sometimes a vector can be represented in matrix form as:

\ LARGE \ vec {V} = \ begin {bmatrix} V_ {1} \\ V_ {2} \\ V_ {3} \\ \ vdots \\ V_ {n} \ end {bmatrix}; \ vec {V} = \ begin {bmatrix} V_ {1}, V_ {2}, V_ {3}, \ cdots, V_ {n} \\ \ end {bmatrix}

Characteristics of a vector

A vector can be defined by its coordinates, if it is in the xy plane, it is represented as follows:
\ LARGE \ vec {V} = (V_ {x}, V_ {y})

Being Vx and Vy its coordinates.

Now, if we consider a right triangle, whose components are its sides and whose hypotenuse has a magnitude V, we find that:

V_ {x} = V \ cos \ theta; V_ {y} = V \ sin \ theta

And with the vector being the vector sum of its coordinates:

\ large \ vec {V} = \ vec {V_ {x}} + \ vec {V_ {y}}

If a vector is three real dimensions, represented on the x, y, and z axes, it can be represented:

\ LARGE \ vec {V} = \ left (V_ {x}, V_ {y}, V_ {z} \ right)

Where Vx, Vy and Vz are its coordinates.

Elements of a vector

If we represent a vector graphically we can differentiate the following elements:

  • The module is the length proportional to the value of the vector.

vector module

  • The direction is the direction of the line that contains the vector or any line parallel to it.

direction vector

  • The sense, indicated by the arrowhead (end), being one of the two possible on the straight support.

sense vector

Vector types

These are some of the vector types:

Null vector

They have a module of zero (0) and no direction or sense can be assigned to them; its end and origin lie at the same point.

Unit Vectors

They have unit (1) as a module. They are commonly denoted with a circumflex accent, "^".

Team or equal vectors

Two vectors are teamlens if they have the same modulus and the same direction and sense.

Opposite vectors

Two vectors are opposite if they have the same modulus and the same direction, but opposite direction. The following vector is the opposite of the previous vectors.

Concurrent or Angular Vectors

They are those whose directions or lines of action pass through the same point. They are also called angular because they form an angle between them.


The three vectors are concurrent.

Collinear Vectors

They are those who share the same line of action, that is, if both are parallel to a line or are on the same line.

Coplanar Vectors

They are those vectors whose lines of action are located in the same plane, that is, if both are parallel to the same plane or are in the same plane.


All three vectors are coplanar.

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