por What is a vector quantity?
A vector magnitude It is any expression represented by a vector that has a numerical value (module), direction, meaning and point of application. Some examples of vector quantities are displacement, velocity, force, and electric field.
The graphic representation of a vector quantity consists of an arrow whose tip indicates its direction and sense, its length is the module and the starting point is the origin or point of application.
The vector quantity is analytically represented by a letter with an arrow at the top pointing to the right in the horizontal direction. It can also be represented by a letter written in bold. V whose module ǀVǀ is written in cursive v.
One of the applications of the concept of vector magnitude 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.
Elements of a vector quantity
A vector magnitude is any entity represented by a line segment, oriented in space, that has the characteristics of a vector. Its elements are:
Module: It is the numerical value that indicates the size or intensity of the vector quantity.
Address: 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.
Sense: Indicated by the arrowhead at the end of the vector.
point of application: 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, equipollent, fixed, and unitary.
Collinear: They belong to or act on the same straight line, they are also called linearly dependent and they can be vertical, horizontal and inclined.
parallels: They have the same direction or inclination.
perpendiculars: two vectors are perpendicular to each other when the angle between them is 90°.
Concurrent: They are vectors that, when sliding on their line of action, coincide at the same point in space.
coplanar: They act in a plane, for example the plane xy.
free: They move at any point in space maintaining their module, direction and meaning.
sliders: They move along the line of action determined by their direction.
opposites: They have the same module and direction, and the opposite sense.
equipolentes: They have the same module, direction and meaning.
Fixed: They have an invariable point of application.
unitary: Vectors whose module is unity.
vector components
A vector magnitude in a three-dimensional space is represented in a system of three mutually perpendicular axes (X and Z) called the orthogonal trihedron.
In the image the vectors vx, vy, vz are the vector components of the vector V whose unit vectors are x,and,z. vector magnitude V It is represented by the sum of its vector components.
V =Vx + vy + vz
The resultant of various vector magnitudes is the vector sum of all vectors and replaces those vectors in a system.
vector field
The vector field is the region of space in which each of its points corresponds to a vector quantity. If the magnitude that is manifested is a force acting on a body or physical system, then the vector field is a force field.
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.
Vector operations
vector addition: It is the resultant of two or more vectors. If we have two vectors EITHER and P the sum is EITHER + P = Q. the vector Q is the resultant vector that is obtained graphically by translating the origin of the vector TO at the end of the vector B..
vector subtraction: The subtraction of two vectors O and P is EITHER – P = Q. The vector Q is obtained by adding to the vector EITHER its opposite –P. The graphical method is the same as the sum with the difference that the opposite vector is moved to the extreme.
Scalar product: The product of a scalar magnitude to by a vector quantity P is a vector mP which has the same direction of the vector Q. If the scalar magnitude is zero, the scalar product is a zero vector.
Examples of vector quantities
Position
The position of an object or particle with respect to a reference system is a vector that is given by its rectangular coordinates X and Zand is represented by its vector components xî, yĵ, zk. vectors Yo, ĵ, what They are unit vectors.
A particle at a point (X and Z) has a position vector r = xî + yĵ + zk. The numerical value of the position vector is r= √(x2 + y2 + z2). The change in position of the particle from one position to another with respect to a reference frame is the vector displacement Δr and is calculated with the following vector expression:
Δr = r2 – r1
Acceleration
The average acceleration (tom) is defined as the variation of velocity v in a time interval Δt and the expression to calculate it is tom=Δv/Δtbeing Δv the vector change of velocity.
The instantaneous acceleration (to) is the limit of the average acceleration tom when Δt becomes so small that it tends to zero. Instantaneous acceleration is expressed as a function of its vector components
to =axi +oh ĵ+ azk
gravitational field
The force of gravitational attraction exerted by a mass mlocated at the origin, on another mass m at a point in space x, and, z is a vector field called the gravitational force field. This force is given by the expression:
F=(-mMG/r)ȓ
r = xî + yĵ + zk
F = is the physical magnitude of the gravitational force
G = is the Universal gravitational constant
ȓ = is the position vector of the mass m
References
Tallack, J C. Introduction to Vector Analysis. Cambridge: Cambridge University Press, 2009.
Spiegel, MR, Lipschutz, S, and Spellman, D. Vector Analysis. sl : Mc Graw Hill, 2009.
Brand, L. Vector Analysis. New York: Dover Publications, 2006.
Griffiths, D J. Introduction to Electrodynamics. New Jersey: Prentice Hall, 1999. pp. 1-10.
Hague, b. An Introduction to Vector Analysis. Glasgow: Methuen & Co. Ltd, 2012.