CONCURRENT VECTORS: CHARACTERISTICS, EXAMPLES AND EXERCISES - MATHS - 2025

The concurrent vectors are vectors groups whose axes coincide at one point, forming between each pair of internal and external another angle. A clear example is seen in the figure below, where A, B and C are vectors concurrent with each other.

D and E unlike the rest are not. There are angles formed between the concurrent vectors AB, AC, and CB. They are called relationship angles between the vectors.

characteristics

-They have a point in common, which coincides with their origin: all the magnitudes of the concurrent vectors start from a common point to their respective ends.

-The origin is considered as the point of action of the vector: an action point must be established which will be directly affected by each of the concurrent vectors.

-Its domain in the plane and space is R ² and R ³ respectively: the concurrent vectors are free to cover the entire geometric space.

-Allows different notations in the same group of vectors. According to the branches of study, different notations are present in operations with vectors.

Types of vectors

The branch of vectors has multiple subdivisions, among some they can be named: parallel, perpendicular, coplanar, corresponding, opposite and unitary. Concurrent vectors are listed here, and like all those named above, they have many applications in different sciences.

They are very common in the study of vectors, because they represent a useful generalization in the operations with them. Both in the plane and in space, concurrent vectors are commonly used to represent different elements and study their influence on a particular system.

Vector notation

There are several ways to represent a vector element. The main and best known are:

Cartesian

Proposed by this same mathematical approach, it denotes the vectors with a triple corresponding to the magnitudes of each axis (x, y, z)

A: (1, 1, -1) Space A: (1, 1) Plane

Polar

They serve only to denote vectors in the plane, although in the integral calculus it is assigned the depth component. It is composed with a linear magnitude r and an angle with respect to the polar axis Ɵ.

A: (3, 45 ⁰) Plane A: (2, 45 ⁰, 3) Space

Analytical

They define the magnitudes of the vector using the versores. The versores (i + j + k) represent the unit vectors corresponding to the axes X, Y and

A: 3i + 2j - 3k

Spherical

They are similar to polar notation, but with the addition of a second angle sweeping over the xy plane symbolized by δ.

A: (4, 60 ^or, π / 4)

Concurrent vector operations

Concurrent vectors are mostly used to define operations between vectors, because it is easier to compare the elements of vectors when they are presented concurrently.

Sum (A + B)

The sum of concurrent vectors aims to find the resultant vector V _r. Which, according to the branch of study, corresponds to a final action

For example: 3 strings {A, B, C} are tied to a box, each end of the string is held by one subject. Each of the 3 subjects must pull the rope in a different direction than the other 2.

A: (ax, ay, az) B: (bx, by, bz) C: (cx, cy, cz)

A + B + C = (ax + bx + cx; ay + by + cy; az + bz + cz) = V _r

The box will only be able to move in one direction, therefore V _r will indicate the direction and direction of the box's movement.

Difference (A - B)

There are many criteria regarding the difference between vectors, many authors choose to exclude it and state that only the sum between vectors is stipulated, where the difference is about the sum of the opposite vector. The truth is that vectors can be subtracted algebraically.

A: (ax, ay, az) B: (bx, by, bz)

A - B = A + (-B) = (ax-bx; ay-by; az-bz) =

Scalar product (A. B)

Also known as a dot product, it generates a scalar value that can be related to various magnitudes depending on the branch of study.

For geometry, indicate the area of ​​the parallelogram formed by the pair of concurrent vectors through the parallelogram method. For mechanical physics it defines the work done by a force F when moving a body a distance Δr.

ѡ = F. Δr

As its name indicates, it generates a scalar value and is defined as follows:

Let the vectors A and B be

A: (ax, ay, az) B: (bx, by, bz)

-Analytical form:

(A. B) = -A -.- B-.Cos θ

Where θ is the internal angle between both vectors

-Algebraic form:

(A. B) = (ax.bx + ay.by + az.bz)

Cross product (A x B)

The vector product or dot product between two vectors, defines a third vector C having the quality of being perpendicular to B and C. In physics, the torque vector τ is the base element of rotational dynamics.

-Analytical form:

- A x B - = -A -.- B-.Sen θ

-Algebraic form:

(A x B) = = (ax. By - ay. Bx) - (ax. Bz - az. Bx) j + (ax. By - ay. Bx) k

-Relative movement: r _{A / B}

The basis of relativity is relative motion and concurrent vectors are the basis of relative motion. Relative positions, velocities and accelerations can be deduced by applying the following order of ideas.

r _{A / B} = r _A - r _B; Relative position of A with respect to B

v _{A / B} = v _A - v _B; Relative velocity of A with respect to B

a _{A / B} = a _A - a _B; Relative acceleration of A with respect to B

Examples: solved exercises

Exercise 1

Let A, B, and C be concurrent vectors.

A = (-1, 3, 5) B = (3, 5, -2) C = (-4, -2, 1)

-Define the resulting vector V _r = 2A - 3B + C

2A = (2 (-1), 2 (3), 2 (5)) = (-2, 6, 10)

-3B = (-3 (3), -3 (5), -3 (-2)) = (-9, -15, 6)

V _r = 2A + (-3B) + C = (-2, 6, 10) + (-9, -15, 6) + (-4, -2, 1)

V _r = (;; (10 + 6 + 1))

V _r = (-15, -11, 17)

-Define the dot product (A. C)

(A. C) = (-1, 3, 5). (-4, -2, 1) = (-1) (-4) + 3 (-2) + 5 (1) = 4 - 6 + 5

(A. C) = 3

-Calculate the angle between A and C

(A. C) = -A -.- C-. Cos θ Where θ is the shortest angle between the vectors

θ = 88.63 ⁰

-Find a vector perpendicular to A and B

For this, it is necessary to define the vector product between (-1, 3, 5) and (3, 5, -2). As explained before, a 3 x 3 matrix is ​​constructed where the first row is composed of the triple unit vectors (i, j, k). Then the 2nd and 3rd rows are made up of the vectors to operate, respecting the operational order.

(A x B) = = i - j + k

(A x B) = (-5 - 9) I - (2 - 15) j + (-5 - 9) k

(A x B) = - 14 I + 13 j - 14 k

Exercise 2

Let V _a and V _{b be} the velocity vectors of A and B respectively. Calculate the velocity of B seen from A.

V _a = (3, -1, 5) V _b = (2, 5, -3)

In this case the relative velocity of B with respect to A V _{B / A} is requested

V _{B / A} = V _B - V _A

V _{B / A} = (2, 5, -3) - (3, -1, 5) = (-1, 6, -8)

This is the velocity vector of B seen from A. Where a new vector of the velocity of B is described taking reference from an observer positioned at A and moving with the velocity of A.

Proposed exercises

1-Construct 3 vectors A, B and C that are concurrent and relate 3 operations between them through a practical exercise.

2-Let the vectors A: (-2, 4, -11), B: (1, -6, 9) and C: (-2, -1, 10). Find vectors perpendicular to: A and B, C and B, The sum A + B + C.

4-Determine 3 vectors that are perpendicular to each other, without taking into account the coordinate axes.

5-Define the work done by a force that lifts a block of mass 5 kg, from the bottom of a well 20m deep.

6-Show algebraically that the subtraction of vectors is equal to the sum of the opposite vector. Justify your postulates.

7-Denote a vector in all the notations developed in this article. (Cartesian, polar, analytic and spherical).

8-The magnetic forces exerted on a magnet that rests on a table, is given by the following vectors; V: (5, 3, -2), T: (4, 7, 9), H: (-3, 5, -4). Determine in which direction the magnet will move if all the magnetic forces act at the same time.

References

1. Euclidean Geometry and Transformations. Clayton W. Dodge. Courier Corporation, Jan 1 2004
2. How to Solve Applied Mathematics Problems L. Moiseiwitsch. Courier Corporation, Apr 10 2013
3. Basic Concepts of Geometry. Walter Prenowitz, Meyer Jordan. Rowman & Littlefield, Oct 4. 2012
4. Vectors. Rocío Navarro Lacoba, Jun 7. 2014
5. Linear algebra. Bernard Kolman, David R. Hill. Pearson Education, 2006