RESULTING VECTOR: CALCULATION, EXAMPLES, EXERCISES - PHYSICAL - 2025

The resulting vector is the one obtained by an operation with vectors whose result is also a vector. Normally this operation is the sum of two or more vectors, by means of which a vector is obtained whose effect is equivalent.

In this way, vectors such as the resulting velocity, acceleration or force are obtained. For example, when several forces F ₁, F ₂, F ₃,… act on a body. the vector sum of all these forces is equal to the net force (the resultant), which is mathematically expressed as follows:

F ₁ + F ₂ + F ₃ +… = F _R or F _N

Figure 1. The weight of the snow is distributed on the roof and its action can be replaced by a single resultant force applied in the appropriate place. Source: Pixabay.

The resulting vector, whether it is forces or any other vector magnitude, is found by applying the rules of vector addition. As the vectors have direction and sense as well as a numerical value, it is not enough to add the modules to have the resulting vector.

This is true only in the case where the vectors involved are in the same direction (see examples). Otherwise, it is necessary to use vector sum methods, which depending on the case can be geometric or analytical.

Examples

Geometric methods for finding the resulting vector are the traverse method and the parallelogram method.

As for analytical methods, there is the component method, by which the vector resulting from any system of vectors can be found, provided that we have its Cartesian components.

Geometric methods to add two vectors

Suppose the vectors u and v (we denote them in bold to distinguish them from the scalars). In figure 2a) we have them located on the plane. In figure 2 b) it has been translated to vector v in such a way that its origin coincides with the end of u. The resulting vector goes from the origin of the first (u) to the tip of the last (v):

Figure 2. The resulting vector from the graphical sum of vectors. Source: self made.

The resulting figure in this case is a triangle (a triangle is a 3-sided polygon). If we have two vectors in the same direction, the procedure is the same: place one of the vectors after the other and draw one that goes from the origin or tail of the first to the tip or end of the last.

Note that the order in which this procedure is done does not matter, since the sum of vectors is commutative.

Also note that in this case the module (the length or size) of the resulting vector is the sum of the modules of the added vectors, unlike the previous case, in which the module of the resulting vector is less than the sum of the participant modules.

Parallelogram method

This method is very appropriate when you need to add two vectors whose origin points coincide, say, with the origin of an xy coordinate system. Suppose this is the case for our vectors u and v (figure 3a):

Figure 3. Sum of two vectors using the parallelogram method with the resulting vector in turquoise blue. Source: self made.

In figure 3b) a parallelogram has been constructed with the help of dotted lines parallel to u and v. The resulting vector has its origin at O ​​and its end at the point where the dotted lines intersect. This procedure is completely equivalent to that described in the preceding section.

Exercises

-Exercise 1

Given the following vectors, find the resulting vector using the traverse method.

Figure 4. Vectors to find their resultant using the polygonal method. Exercise 1. Source: own elaboration.

Solution

The traverse method is the first of the methods seen. Remember that the sum of vectors is commutative (the order of the addends does not alter the sum), so you can start with any of the vectors, for example u (figure 5a) or r (figure 5b):

Figure 5. Sum of vectors using the polygonal method. Source: self made.

Figure obtained is a polygon and the resulting vector (in blue) is called R. If you start with another vector, the shape that is formed may be different, as shown in the example, but the resulting vector is the same.

Exercise 2

In the following figure we know that the modules of the vectors u and v respectively are u = 3 arbitrary units and v = 1.8 arbitrary units. The angle that u makes with the positive x-axis is 45º, while v makes 60º with the y-axis, as seen in the figure. Find the resultant vector, magnitude, and direction.

Solution

In the preceding section the resulting vector was found by applying the parallelogram method (in turquoise in the figure).

An easy way to find the resulting vector analytically is to express the addend vectors in terms of their Cartesian components, which is an easy task when modulus and angle are known, such as the vectors in this example:

u _x = u. cos 45º = 3 x cos 45º = 2.12; u _y = u. sin 45º = 3x sin 45º = 2.12

v _x = v. sin 60º = 1.8 x sin 60º = 1.56; v _y = -v. cos 60º = -1.8 x cos 60º = - 0.9

The vectors u and v are vectors belonging to the plane, therefore having two components each. Vector u is in the first quadrant and its components are positive, while vector v is in the fourth quadrant; its x component is positive, but its projection on the vertical axis falls on the negative y axis.

Calculation of the Cartesian components of the resultant vector

The resulting vector is found by adding algebraically the respective x and y components, to obtain their Cartesian components:

R _x = 2.12 + 1.56 = 3.68

R _y = 2.12 + (-0.9) = 1.22

Once the Cartesian components have been specified, the vector is fully known. The resulting vector can be expressed with the notation in brackets:

R = <3.68; 1.22> arbitrary units

Bracket notation is used to distinguish a vector from a point in the plane (or in space). Another way to express the resulting vector analytically is by using the unit vectors i and j in the plane (i, j and k in space):

R = 3.68 i + 1.22 j arbitrary units

Since both components of the resulting vector are positive, vector R belongs to the first quadrant, which has already been seen graphically before.

Magnitude and direction of the resultant vector

Knowing the Cartesian components, the magnitude of R is calculated through the Pythagorean theorem, since the resulting vector R, together with its components R _x and R _and form a right triangle:

Magnitude or module: R = (3.68 ² + 1.22 ²) ^½ = 3.88

Direction q taking the positive x axis as a reference: q = arctan (R _y / R _x) = arctg (1.22 /3.68) = 18.3 º

References

1. Adding Vectors and Rules. Retrieved from: newt.phys.unsw.edu.au
2. Figueroa, D. Series: Physics for Sciences and Engineering. Volume 1. Kinematics. 31-68.
3. Physical. Module 8: Vectors. Recovered from: frtl.utn.edu.ar
4. Hibbeler, R. 2006. Mechanics for Engineers. Static 6th Edition. Continental Publishing Company. 15-53.
5. Vector Addition Calculator. Retrieved from: www.1728.org