VECTOR MAGNITUDE: WHAT IT CONSISTS OF AND EXAMPLES - PHYSICAL - 2025

A vector quantity is any expression represented by a vector that has a numerical value (modulus), direction, direction and point of application. Some examples of vector quantities are displacement, velocity, force, and the electric field.

The graphic representation of a vector quantity consists of an arrow whose tip indicates its direction and direction, its length is the module and the starting point is the origin or point of application.

Graphic representation of a vector

The vector quantity is represented analytically by a letter bearing an arrow at the top pointing to the right in a horizontal direction. It can also be represented by a bold letter V whose modulus ǀ V ǀ is written in italics V.

One of the applications of the vector magnitude concept is in the design of highways and roads, specifically in the design of their curvatures. Another application is the calculation of the displacement between two places or the change of speed of a vehicle.

What is a vector quantity?

A vector quantity is any entity represented by a line segment, oriented in space, that has the characteristics of a vector. These characteristics are:

Modulus: It is the numerical value that indicates the size or intensity of the vector magnitude.

Direction: It is the orientation of the line segment in the space that contains it. The vector can have a horizontal, vertical or inclined direction; north, south, east, or west; northeast, southeast, southwest, or northwest.

Direction: Indicated by the arrowhead at the end of the vector.

Application point: It is the origin or initial actuation point of the vector.

Vector classification

Vectors are classified as collinear, parallel, perpendicular, concurrent, coplanar, free, sliding, opposite, team-lens, fixed, and unit.

Collinear: They belong or act on the same straight line, they are also called linearly dependent and can be vertical, horizontal and inclined.

Parallel: They have the same direction or inclination.

Perpendicular - Two vectors are perpendicular to each other when the angle between them is 90 °.

Concurrent: They are vectors that when sliding along their line of action coincide at the same point in space.

Coplanaries: They act in a plane, for example the xy plane.

Free: They move at any point in space, keeping their module, direction and sense.

Sliders: They move along the line of action determined by their direction.

Opposites: They have the same module and direction, and the opposite direction.

Equipolentes: They have the same module, direction and sense.

Fixed: They have the point of application invariable.

Unitary: Vectors whose module is the unit.

Vector components

A vector quantity in a three-dimensional space is represented in a system of three mutually perpendicular axes (x, y, z) called an orthogonal trihedron.

Vector components of a vector magnitude. from Wikimedia Commons

In the image the vectors Vx, Vy, Vz are the vector components of the vector V whose unit vectors are x, y, z. The vector magnitude V is represented by the sum of its vector components.

The resultant of several vector quantities is the vector sum of all vectors and replaces these vectors in a system.

Vector field

The vector field is the region of space in which a vector magnitude corresponds to each of its points. If the magnitude that is manifested is a force acting on a body or physical system then the vector field is a field of forces.

The vector field is represented graphically by field lines that are tangent lines of the vector magnitude at all points in the region. Some examples of vector fields are the electric field created by a point electric charge in space and the velocity field of a fluid.

Electric field created by a positive electric charge.

Vector operations

Acceleration

The mean acceleration (a _m) is defined as the variation of the velocity v in a time interval Δt and the expression to calculate it is a _m = Δv / Δt, where Δv is the speed change vector.

The instantaneous acceleration (a) is the limit of the mean acceleration at _m when Δt becomes so small that it tends to zero. Instantaneous acceleration is expressed as a function of its vector components

Gravitational field

The gravitational attractive force exerted by a mass M, located at the origin, on another mass m at a point in x, y, z space is a vector field called the gravitational force field. This force is given by the expression:

References

1. Tallack, J C. Introduction to Vector Analysis. Cambridge: Cambridge University Press, 2009.
2. Spiegel, MR, Lipschutz, S and Spellman, D. Vector Analysis. sl: Mc Graw Hill, 2009.
3. Brand, L. Vector Analysis. New York: Dover Publications, 2006.
4. Griffiths, D J. Introduction to Electrodynamics. New Jersey: Prentice Hall, 1999. pp. 1-10.
5. Hague, B. An Introduction to Vector Analysis. Glasgow: Methuen & Co. Ltd, 2012.