vector definition

Evelyn Maitee Marin
Industrial Engineer, MSc in Physics, and EdD

A vector is a standard element of visual property symbolized from an oriented line segment (arrow), whose function, in Physics, allows to present quantities that have magnitude, direction and meaning, as is the case of displacement, velocity, acceleration, force, static momentum, angular momentum, among other quantities. However, vectors are exclusive to Physics, since in other areas such as Algebra and Geometry, they are also applied, although generally in a slightly more abstract way.

To graph the vectors, the use of some coordinate system is required as a reference to use the scale to represent its magnitude, direction and sense. In most cases, the Cartesian coordinate system is used, but other systems such as polar, cylindrical or spherical coordinates can also be used.

Force is a vector physical quantity

Vector characteristics

As mentioned, vectors are represented by an arrow that has a start, an end, a line of action, and a size:

Characteristics of a vector

Magnitude: This characteristic is also called the norm or modulus, and represents the size of the vector. This property is of great value to represent other physical quantities, since even when measuring (for example, with a ruler), the size of the vector the dimension of that measurement is length, an equivalent can be defined between that length and the magnitude of the vector quantity to which it is associated; say, 10 cm → 20 N.

Direction: this characteristic expresses the inclination of the vector with respect to a reference axis (usually a Cartesian axis) and is expressed by an angle. When it is desired to graphically represent or measure the direction of the vector, a protractor is used. Many authors usually express the direction of a vector in the plane from the positive horizontal axis counterclockwise.

Direction: indicates the orientation of the vector, that is, where the arrow points. For example, A vector can have a vertical direction, but the direction specifies whether it is up or down. Analytically, when the vector is expressed from its rectangular components, the sense is determined by the sense of its components.

Vector classification

There are many criteria for classifying vectors, including their relationship to other vectors or according to a reference:

• According to vector algebra, vectors can be:

o Parallels: they have the same direction and sense (not necessarily the same magnitude).

o Opposites: They have the same magnitude and direction but in the opposite direction.

o Antiparallel: they have the same direction, but the opposite direction (not necessarily the same magnitude).

o Identical: they are vectors that have the same magnitude, direction and meaning. Analytically, this condition implies that the vectors are equal in all their components.

• According to its reference properties:

o Free: they are vectors that are not linked to a reference system, so that they can be placed in any spatial location and will have the same meaning as long as it retains its magnitude, direction and meaning. For example, the static moment vector is equivalent to a couple of forces, which behave as a free vector.

o Fixed: they are vectors that change their spatial location, modifying the effect they produce on the system. These vectors are linked to a reference frame.

By changing the location of the forces, the effect on the deformable body is modified, since the effect of compression is different from that of traction.

Sliding: they are vectors that can be moved only along their line of action. For example, when forces act on rigid bodies, they behave as sliding vectors, which means that regardless of where they are represented along their line of action, they will always have the same effect and result in the system.

According to the reference, the vectors can be fixed, free and sliding.

• According to its value or application:

o Null vector: it is a vector whose magnitude is zero and with arbitrary direction and sense. The definition of this vector arises from the need to justify the subtraction between two identical vectors, since by definition, the result of the subtraction of two vectors must also be a vector.

o Unit vector: it is a dimensionless vector whose module is equal to the unit. This vector is very useful in vector operations to express other vectors, since when multiplying a scalar by a unit vector, this gives it the direction and sense so that the result is a vector with the magnitude of the scalar and the direction and unitary sense.

o Position vector: it is a vector that is used to express the location of a point in space. It goes from the origin of the reference system to the location point of the particle.