Vector magnitude what it consists of and examples

A vector magnitude 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..

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

Article index

• 1 What is a vector magnitude?
  • 1.1 Classification of vectors
  • 1.2 Vector components
  • 1.3 Vector field
  • 1.4 Operations with vectors
• 2 Examples of vector quantities
  • 2.1 Position
  • 2.2 Acceleration
  • 2.3 Gravitational field
• 3 References

What is a vector magnitude?

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

Module: 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.

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, 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 on a plane, for example the plane xy.

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.

Teamlenses: 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 three-dimensional space is represented in a system of three mutually perpendicular axes (X and Z) called orthogonal trihedral.

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.

V = Vx + Vy + Vz

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.

Vector operations

Adding vectors: It is the resultant of two or more vectors. If we have two vectors OR Y P the sum is OR + P = Q. The vector Q is the resulting vector that is obtained graphically by translating the origin of the vector TO to the end of the vector B.

Vector subtraction: The subtraction of two vectors O and P it is OR - P = Q. The vector Q  is obtained by adding to the vector OR its opposite -P. The graphical method is the same as the sum with the difference that the opposite vector is transferred to the extreme.

Scalar product: The product of a scalar quantity to by a vector magnitude P it's a vector mP which has the same direction of the vector P. 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 Z, and is represented by its vector components xî, andĵ, zk. Vectors  î, ĵ, k they are unit vectors.

A particle at a point (X and Z) has a position vector r = xî + and + zk. The numerical value of the position vector is r= √ (x^two + Y^two + z^two). 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 = r_two - r₁

Acceleration

Average acceleration (to_m) is defined as the change in velocity v in a time interval Δt and the expression to calculate it is to_m= Δv / Δt, being Δv the speed change vector.

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

to =to_xî +to_Y ĵ+ to_zk

Gravitational field

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

F= (- mMG /r)ȓ

r = xî + andĵ + zk

F = is the physical magnitude gravitational force

G = is the Universal gravitation constant

ȓ = is the position vector of the mass m

References

1. Tallack, J C. Introduction to Vector Analysis. Cambridge: Cambridge University Press, 2009.
2. Spiegel, M R, Lipschutz, S and Spellman, D. Vector Analysis. s.l. : 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.