Vector Magnitude: What It Is and Examples

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

The graphical representation of a vector magnitude consists of an arrow whose head indicates its direction and direction, its length is the magnitude and the starting point is the origin or application point.

The magnitude of the vector is analytically represented with a letter that has 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 ǀV ǀ 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 displacement between two locations or the change of speed of a vehicle.

What is a vector magnitude?

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

Modulus : is the numerical value that indicates the size or magnitude of the vector’s magnitude.

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

Direction : Is indicated with the arrowhead at the end of the vector.

Application point : is the origin or initial action point of the vector.

vector classification

Vectors are classified into collinear, parallel, perpendicular, concurrent, coplanar, free, sliding, opposite, equal, fixed, and unitary.

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

Parallels : They have the same direction or slope.

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

Concurrent : they are vectors that, when sliding over their line of action, coincide at the same point in space.

Coplanars : They act on a plane, for example, the xy plane.

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

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

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

Equivalent : They have the same module, direction and meaning.

Fixed : They invariably have the application point.

Unitary : vectors whose modulus is unity.

vector components

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

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

= Vx + Vy + Vz

The result of several quantities of vectors 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 a vector magnitude corresponds to each of its points. If the magnitude that manifests is a force acting on a physical body or system, then the vector field is a force field.

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

vector operations

Adding vectors : is the result of two or more vectors. If there are two vectors O and P the sum is S + P = Q . The vector Q is the resultant vector obtained graphically by moving the origin of the vector The parafinal the vector B .

Vector Subtraction : The subtraction of two vectors of O and P is O – P = The vector Q. Q obtained is the addition of the vector or its opposite – P The graphical method is the same as summation except that the opposite vector is moved to the extreme.

Product climb : The product of a magnitude climb to by a magnitude of vector P is a vector vP which has the same direction as the vector P. If the magnitude scale is zero, the dot product is a zero vector.

Examples of vector quantities

Position

The position of an object or particle relative to a frame of reference is a vector that is given by its rectangular coordinates x, y, z and is represented by its vector components  ,  , zk . The vectors î , ĵ , k are unit vectors.

A particle at a point ( x, y, z ) has a position vector r =  +  + zk . The numerical value of the position vector is r = √ ( 2 + y 2 + z 2 ). The change in position of the particle from one position to another relative to a reference system is the displacement vector Δr and is calculated with the following vector expression:

Δr = r 2 – r 1

Acceleration

The average acceleration ( m ) is defined as the change in velocity v over a time interval .DELTA.t and expression to calculate is m = dv / dt , where Dv vector rate change.

The instantaneous acceleration ( a ) is the limit of the average acceleration for m when Δt becomes so small that it tends to zero. The instantaneous acceleration is expressed based on its vector components.

a = x î + y ĵ + z k

gravitational field

The gravitational attraction 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 strength is given by the expression:

F = (-mMG / r ) ȓ

r =  +  + zk

F = is the gravitational force of physical magnitude

G = is the universal gravitation constant

ȓ = is the position vector of mass m

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