Comprehensive Notes on Vectors and Systems of Forces

Fundamental Definition and Elements of Vectors

A vector is a physical quantity that is defined by three essential characteristics: magnitude, direction, and sense. In visual representations, a vector is commonly depicted as an arrow. The length of this arrow corresponds to the magnitude of the quantity, representing how large or intense the force or displacement is. The direction is indicated by the inclination of the arrow, often measured as an angle (for example, $30^{\circ}$) relative to a reference axis such as the x-axis. Finally, the sense of the vector is denoted by the tip of the arrow, indicating the specific orientation toward which the quantity is acting, such as up, down, left, right, or any other point in space.

Two vectors are considered strictly equal if and only if their magnitude, direction, and sense are identical. This equality remains true regardless of where the vectors are positioned in space; as long as these three specific parameters match, the vectors are equivalent. This sets vectors apart from scalar quantities, which are defined solely by their magnitude.

Classification and Systems of Vectors

Vectors are organized into systems based on how their lines of action interact within a physical space. Coplanar vectors are those that exist within a single common plane, whereas non-coplanar vectors are distributed across different planes. A system of collinear vectors occurs when two or more vectors share the same direction or act along the exact same line of action. By contrast, concurrent vector systems are characterized by vectors whose lines of action intersect at a single common point.

Furthermore, force systems can be categorized by the orientation of their supports. In a parallel force system, the lines of action of the various forces do not intersect and remain equidistant from one another. In a collinear system, all involved forces possess the same geometric line of action, effectively pulling or pushing along the same path. In concurrent systems, the distinguishing feature is that all lines of action meet at a shared point of origin or intersection.

Concepts of Resultant and Equilibrant Vectors

The resultant vector is defined as the single vector that represents the combined effect of two or more individual vectors acting simultaneously on a body. It essentially simplifies a complex system into a single "total push" or "total pull" that the object experiences. The resultant is obtained through vector addition, either graphically or algebraically, and its properties are determined by the specific magnitudes, directions, and senses of the original contributing vectors.

Conversely, the equilibrant vector is the vector that perfectly nullifies or cancels out the effect of the resultant vector. It possesses the exact same magnitude as the resultant but acts in the diametrically opposite sense. In practical terms, if a resultant force is pushing an object to the right with a certain intensity, the equilibrant force pushes with equal intensity to the left, resulting in a state of equilibrium. This concept is fundamental in statics and mechanics when studying objects that are at rest or moving with a constant velocity, as the sum of forces including the equilibrant would equal zero.

Vector Addition and Decomposition Methods

The method for adding vectors depends on their spatial relationship. For collinear vectors, the rule is straightforward: if they act in the same sense, their magnitudes are added together; if they act in opposite senses, the smaller magnitude is subtracted from the larger one. For concurrent vectors, several analytical and geometric methods are used. These include the analytical method for vectors with specific angles between them, the parallelogram method for graphical or trigonometric solutions, and the Pythagorean theorem specifically when vectors are perpendicular to one another.

When dealing with an oblique or diagonal vector, it is often necessary to perform vector decomposition. This process breaks a single vector into two perpendicular components: one acting along the horizontal x-axis ($F_x$) and one acting along the vertical y-axis ($F_y$). These components are calculated using the following trigonometric formulas:

$F_x = F \, \cos(\theta)$

$F_y = F \, \sin(\theta)$

Calculating Vector Magnitudes and Unit Vectors

To determine the total magnitude of a vector when its individual components are known, the Pythagorean theorem is applied. For a vector in a two-dimensional plane, the magnitude is calculated as:

$\text{Magnitude} = \sqrt{F_x^2 + F_y^2}$

For a vector acting in three-dimensional space with three components ($i, j, k$), the formula expands to:

$\text{Magnitude} = \sqrt{x^2 + y^2 + z^2}$

A unit vector is a specific type of vector that preserves the direction and sense of a given vector but has a normalized magnitude exactly equal to 1. To calculate a unit vector, each individual component of the original vector must be divided by the vector's total magnitude. For example, given a vector $\mathbf{A} = -2\mathbf{i} + 4\mathbf{j}$, the magnitude is first found as $\sqrt{(-2)^2 + 4^2} = \sqrt{20}$. The unit vector is then expressed as:

$\mathbf{u} = (\frac{-2}{\sqrt{20}})\mathbf{i} + (\frac{4}{\sqrt{20}})\mathbf{j}$

Mechanical Centers of a Rigid Body

In the study of rigid bodies, two distinct centers are identified. The centroid is defined as the strictly geometric center of an object. The center of gravity, however, is the specific point where the entire effect of gravity on a rigid body is concentrated. While they may coincide in uniform objects, they represent different physical concepts: one being purely spatial/geometric and the other being related to mass and gravitational force distribution.

Questions & Discussion

Reactivo 1: In a matching exercise relating force systems to their characteristics, the following pairings were established. Parallel systems are those where the supports of the forces are parallel (1A). Collinear systems are defined as those where all forces possess the same line of action (2C). Concurrent systems consist of forces where the lines of action share a common point (3B). The correct sequence is 1A, 2C, 3B.

Reactivo 2: A particle undergoes three displacements: $\mathbf{r}_1 = -2\mathbf{i}+5\mathbf{j}+3\mathbf{k}$, $\mathbf{r}_2 = 4\mathbf{i}-3\mathbf{j}+4\mathbf{k}$, and $\mathbf{r}_3 = -\mathbf{i}+2\mathbf{j}+\mathbf{k}$. To find the magnitude of the resulting displacement, the components must first be summed: $\mathbf{R}_x = -2 + 4 - 1 = 1$, $\mathbf{R}_y = 5 - 3 + 2 = 4$, and $\mathbf{R}_z = 3 + 4 + 1 = 8$. The magnitude is then calculated as $\sqrt{1^2 + 4^2 + 8^2} = \sqrt{1 + 16 + 64} = \sqrt{81} = 9$. The correct answer is 9.

Reactivo 3: Vectors are defined as magnitudes possessed of a modulus (magnitude), direction, and sense, which are summed in the same manner as displacements. The correct missing term is direction.

Reactivo 4: The point that defines the geometric center of an object is known as the centroid.

Reactivo 5: To calculate the magnitude of the force vector $\mathbf{F} = (-2\mathbf{i}-2\mathbf{j}+\mathbf{k}) \, \text{N}$, the formula is $\sqrt{(-2)^2 + (-2)^2 + (1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3\, \text{N}$. The magnitude is 3N.

Reactivo 6: A person walks for four hours: 4 km north in the first hour, 3 km east in the second, 4 km south in the third, and 3 km west in the last. Because the northward movement is cancelled by the southward movement ($4 - 4 = 0$) and the eastward movement is cancelled by the westward movement ($3 - 3 = 0$), the total displacement magnitude is 0 km.

Reactivo 8: The point where the effect of gravity is concentrated in a rigid body is the center of gravity.

Reactivo 9: To find the magnitude of the force that nullifies the resultant of two forces, 5 N (north) and 10 N (west), we first find the resultant magnitude. Because these are perpendicular, $\sqrt{5^2 + 10^2} = \sqrt{25 + 100} = \sqrt{125}\, \text{N}$. Since the equilibrant has the same magnitude as the resultant, the answer is $\sqrt{125}\, \text{N}$.

Reactivo 10: A passenger moves 6 m south on a train platform while the car moves 9 m west. The resultant displacement in meters is calculated using the Pythagorean theorem: $\sqrt{6^2 + 9^2} = \sqrt{36 + 81} = \sqrt{117} \approx 10.81\, \text{m}$. The correct answer is 10.81 m.

Reactivo 12: The force of equilibrium is equal in magnitude to the resultant force but acting in the opposite sense.

Reactivo 13: A boat moves at $2\, \text{m/s}$ and crosses a river current of $2\, \text{m/s}$ perpendicularly. The resultant velocity magnitude is $\sqrt{2^2 + 2^2} = \sqrt{8}\, \text{m/s}$. The angle is found by $\tan^{-1}(\frac{2}{2}) = \tan^{-1}(1)$, which is $45^{\circ}$. The result is $\sqrt{8}\, \text{m/s}, 45^{\circ}$.

Reactivo 14: The magnitude of the vector $\mathbf{F} = 3\mathbf{i} + 4\mathbf{j}$ is $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

Reactivo 15: Two forces act on a body: $F_1 = 60\, \text{N}$ at $0^{\circ}$ and $F_2 = 80\, \text{N}$ at $90^{\circ}$. Since they are perpendicular, the magnitude of the resultant is $\sqrt{60^2 + 80^2} = \sqrt{3600 + 6400} = \sqrt{10000} = 100\, \text{N}$.

Reactivo 17: When considering the vectorial nature of two perpendicular forces, the magnitude of the resultant force is always less than the simple algebraic sum of the magnitudes of the component forces. This is a consequence of the triangle inequality, as the hypotenuse of a right triangle is shorter than the sum of the two legs.