TEAMLENS VECTORS: DEFINITION, NOTATION, EXERCISES - PHYSICAL - 2025

Two or more vectors are Equipolentes if they have the same module, the same direction and the same sense, even when their point of origin is different. Remember that the characteristics of a vector are precisely: origin, module, direction and sense.

Vectors are represented by an oriented segment or arrow. Figure 1 shows the representation of several vectors in the plane, some of which are team-lensing according to the definition initially given.

Figure 1. Team-lens and non-team-lens vectors. Source: self made.

At a first glance it is possible to see that the three green vectors have the same size, the same direction and the same sense. The same can be said about the two pink vectors and the four black vectors.

Many magnitudes of nature have a vector-like behavior, such is the case of speed, acceleration, and force, to name just a few. Hence the importance of properly characterizing them.

Notation for vectors and equipment

To distinguish vector quantities from scalar quantities, bold typeface or an arrow over the letter is often used. When working with vectors by hand, on the notebook, it is necessary to distinguish them with the arrow and when using a printed medium, bold type is used.

Vectors can be denoted by indicating their point of departure or origin and their point of arrival. For example AB, BC, DE and EF in figure 1 are vectors, while AB, BC, DE and EF are scalar quantities or numbers that indicate the magnitude, modulus or size of their respective vectors.

To indicate that two vectors are team-oriented, the symbol « ∼« is used. With this notation, in the figure we can point out the following vectors that are team-oriented to each other:

AB∼BC∼DE∼EF

They all have the same magnitude, direction and meaning. Therefore, they comply with the regulations indicated above.

Free, sliding and opposite vectors

Any of the vectors in the figure (for example AB) is representative of the set of all equipment-lens fixed vectors. This infinite set defines the class of free vectors u.

u = { AB, BC, DE, EF,….. }

An alternative notation is the following:

If the boldface or the little arrow is not placed above the letter u, it means that we want to refer to the module of the vector u.

The free vectors are not applied to any particular point.

On the other hand, the sliding vectors are team-resistant vectors to a given vector, but their point of application must be contained in the line of action of the given vector.

And opposite vectors are vectors that have the same magnitude and direction but opposite senses, although in English texts they are called opposite directions since the direction also indicates the direction. The opposite vectors are not team-oriented.

Exercises

-Exercise 1

Which other vectors than those shown in Figure 1 are team-leaning to each other?

Solution

Apart from those already indicated in the previous section, it can be seen from figure 1 that AD, BE and CE are also team-friendly vectors:

AD ∼ BE ∼ CE

Any of them is representative of the class of free vectors v.

The vectors AE and BF are also team-lensing:

AE ∼ BF

Which are representatives of class w.

-Exercise 2

Points A, B and C are on the Cartesian plane XY and their coordinates are:

A = (- 4.1), B = (- 1.4) and C = (- 4, -3)

Find the coordinates of a fourth point D such that vectors AB and CD are team-lensing.

Solution

For CD to be team-friendly to AB it must have the same module and the same address as AB.

The modulus of AB squared is:

- AB - ^ 2 = (-1 - (-4)) ^ 2 + (4 -1) ^ 2 = 9 + 9 = 18

The coordinates of D are unknown so we can say: D = (x, y)

Then: - CD - ^ 2 = (x - (- 4)) ^ 2 + (y - (-3)) ^ 2

Since - AB - = - CD - is one of the conditions for AB and CD to be team-lensing, we have:

(x + 4) ^ 2 + (y + 3) ^ 2 = 18

Since we have two unknowns, another equation is required, which can be obtained from the condition that AB and CD are parallel and in the same sense.

Slope of vector AB

The slope of vector AB indicates its direction:

Slope AB = (4 -1) / (- 1 - (-4)) = 3/3 = 1

Indicating that the vector AB forms 45º with the X axis.

Vector CD slope

The slope of CD is calculated in a similar way:

Slope CD = (y - (-3)) / (x - (- 4)) = (y + 3) / (x + 4)

Equating this result with the slope of AB, the following equation is obtained:

y + 3 = x + 4

Which means that y = x + 1.

If this result is substituted in the equation for the equality of the modules, we have:

(x + 4) ^ 2 + (x + 1 + 3) ^ 2 = 18

Simplifying it remains:

2 (x + 4) ^ 2 = 18, Which is equivalent to:

(x + 4) ^ 2 = 9

That is, x + 4 = 3 which implies that x = -1. So the coordinates of D are (-1, 0).

check

The components of vector AB are (-1 - (- 4), 4 -1) = (3, 3)

and those of the CD vector are (-1 - (- 4)); 0 - (- 3)) = (3, 3)

Which means that the vectors are team-oriented. If two vectors have the same Cartesian components, they have the same module and direction, therefore they are team-oriented.

-Exercise 3

The free vector u has magnitude 5 and direction 143.1301º.

Find its Cartesian components and determine the coordinates of points B and C knowing that the fixed vectors AB and CD are team-oriented to u. The coordinates of A are (0, 0) and the coordinates of point C are (-3,2).

Solution

1. Calculation.cc. Fixed vector. Free vector. Recovered from: calculo.cc
2. Descartes 2d. Fixed Vectors and Free Plane Vectors. Recovered from: recursostic.educacion.es
3. Guao project. Vectors teamlenses. Recovered from: guao.org
4. Resnick, R., Krane, K. (2001). Physics. New York: John Wiley & Sons.
5. Serway, R.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks / Cole.
6. Tipler, Paul A. (2000). Physics for Science and Technology. Volume I. Barcelona: Ed. Reverté.
7. Weisstein, E. "Vector." In Weisstein, Eric W. MathWorld. Wolfram Research.