Name: vector components and direction
Uploaded: 2025-11-26T05:44:17.000Z
Duration: 1 min 30 s
Channel: JoVE

Los vectores poseen tanto magnitud como dirección. La magnitud indica la fuerza, mientras que la dirección especifica la orientación en el espacio. Por ejemplo, la velocidad de un automóvil de 50 km / h hacia el este es un vector que define su velocidad y dirección de movimiento.Los vectores se representan como segmentos de línea dirigidos desde un punto inicial hasta un punto terminal, indicados con letras en negrita o una flecha en la parte superior. También se usa comúnmente una media flecha en la parte superior; Ambas convenciones son válidas.Dos vectores se consideran iguales cuando tienen la misma magnitud, lo que indica fuerza, y la misma dirección.Agregar vectores implica colocar el punto inicial de un vector en el punto terminal del otro para formar el resultante. Alternativamente, si ambos vectores se originan en el mismo punto, su suma está representada por la diagonal del paralelogramo que forman, comenzando desde ese mismo punto.Restar vectores implica sumar el negativo de un vector, lo que invierte la dirección del vector. El resultado se muestra mediante la diagonal del paralelogramo, que va desde el punto terminal del vector sustraído hasta el punto terminal del otro vector.En el plano de coordenadas, los vectores se representan como pares ordenados de componentes horizontales y verticales, normalmente encerrados entre corchetes angulares.

vectores

Vectors

Los vectores son entidades matem&#225;ticas que se caracterizan por su magnitud y por su direcci&#243;n. A diferencia de los escalares, definidos &#250;nicamente por su magnitud, los vectores representan magnitudes como el desplazamiento, la velocidad y la fuerza, donde la direcci&#243;n es esencial. Los vectores se representan gr&#225;ficamente como segmentos de recta dirigidos, que se extienden desde un punto inicial hasta un punto terminal, indicados con letras en negrita o con una flecha sobre el s&#237;mbolo. Dos vectores se consideran iguales si tienen la misma magnitud y la misma direcci&#243;n, con independencia de sus posiciones iniciales.Sumar vectores implica colocar el punto inicial de un vector en el punto terminal de otro. Alternativamente, si los vectores parten del mismo punto, su suma es la diagonal del paralelogramo formado. Restar vectores implica sumar el negativo del segundo vector. Geom&#233;tricamente, la diferencia forma la diagonal opuesta del paralelogramo.Multiplicar un vector por un escalar afecta su magnitud, alarg&#225;ndola o reduci&#233;ndola. Un escalar negativo invierte la direcci&#243;n del vector. Por ejemplo, si el vector v se multiplica por &#8722;2, el vector resultante apunta en la direcci&#243;n opuesta y tiene el doble de longitud.Los vectores se representan como pares ordenados de sus componentes horizontal y vertical. Si un vector v se representa en el plano con punto inicial A&#12296;x_1, y_1&#12297; y punto terminal B&#12296;x_2, y_2&#12297;, entonces el vector v puede escribirse como:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:90.18%;'><img style='aspect-ratio:1920/80;' src='https://cloudfront.jove.com/files/ftp_upload/40084/40084_Equation_1_editor_40084.svg' alt='40084_Equation_1.svg' /></figure>La magnitud de un vector v = &#12296;a, b&#12297; se obtiene mediante el teorema de Pit&#225;goras:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:60.45%;'><img style='aspect-ratio:1920/91;' src='https://cloudfront.jove.com/files/ftp_upload/40084/40084_Equation_2_editor_40084.svg' alt='40084_Equation_2.svg' /></figure>Los vectores tambi&#233;n pueden expresarse como combinaciones lineales de vectores unitarios. Un vector unitario es un vector de longitud uno, generalmente denotado como i para la componente horizontal y j para la componente vertical. Un vector v puede escribirse como:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:90.42%;'><img style='aspect-ratio:1920/75;' src='https://cloudfront.jove.com/files/ftp_upload/40084/40084_Equation_3_editor_40084.svg' alt='40084_Equation_3.svg' /></figure>Aqu&#237;, a y b son las componentes horizontal y vertical, respectivamente. Estas componentes tambi&#233;n pueden derivarse de la magnitud y del &#225;ngulo de direcci&#243;n &#952; del vector:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:60.06%;'><img style='aspect-ratio:1920/219;' src='https://cloudfront.jove.com/files/ftp_upload/40084/40084_Equation_4_editor_40084.svg' alt='40084_Equation_4.svg' /></figure>Esta representaci&#243;n simplifica los c&#225;lculos que implican suma, resta y multiplicaci&#243;n escalar de vectores, especialmente al trabajar con componentes vectoriales en el plano cartesiano.

Vectores

Los vectores son entidades matem&#225;ticas que se caracterizan por su magnitud y por su direcci&#243;n. A diferencia de los escalares, definidos ...

Education

Educación

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vector components and direction

Vectors describe quantities with both magnitude and direction. This makes them different from scalars, which have magnitude only. Common examples include displacement, velocity, and force, where direction matters.

A vector is shown as a directed line segment from an initial point to a terminal point. It may also be written with bold letters or with an arrow above the symbol. Two vectors are equal if they have the same magnitude and direction, even when they start in different places.

Vector addition uses tip-to-tail placement. One vector is moved so its initial point begins at the terminal point of another vector. If both vectors start at the same point, their sum is the diagonal of the parallelogram they form.

Vector subtraction is done by adding the negative of the second vector. Geometrically, the difference is the opposite diagonal of the same parallelogram. Multiplying a vector by a scalar changes its length, so the vector can stretch or shrink. A negative scalar also reverses the direction. For example, multiplying vector v by -2 makes the new vector point the opposite way and makes it twice as long.

Vectors can be written as ordered pairs of horizontal and vertical components. If a vector has initial point A⟨x1, y1⟩ and terminal point B⟨x2, y2⟩, it can be expressed from those coordinates. The magnitude of v = ⟨a, b⟩ is found with the Pythagorean theorem.

Vectors can also be written with unit vectors. A unit vector has length one. The standard unit vectors are i for the horizontal direction and j for the vertical direction. In this form, v = ai + bj, where a and b are the horizontal and vertical components.

Those components can also be found from the vector’s magnitude and direction angle θ. This component form makes vector addition, subtraction, and scalar multiplication easier in the Cartesian plane. It is especially useful when working with vectors through their horizontal and vertical parts.

Vector Components and Direction

polar coordinates

Plotting Points with Polar Coordinates

Polar coordinates give a way to plot points in a plane using distance and angle. Instead of x and y values, this system uses a radial distance and an angular direction. It is especially useful for problems with circular or rotational symmetry, including waves, oscillations, and orbital paths in physics and engineering. A point in polar form is written as P(r, θ). The value r is the distance from the pole, which plays the same role as the origin in the Cartesian plane. The angle θ is measured...

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Polar Curves: Circles, Roses, and Limacons

Polar equations can be used to graph curves with radial symmetry or repeating patterns. In the polar coordinate system, a point is described by its distance from the pole, or central point, and by an angle from the polar axis, or reference direction. This format is different from rectangular coordinates and is especially useful for certain curve shapes. A circle centered at the pole appears when the radius stays constant as the angle changes. A line through the pole appears when the angle...

vectors

Vectors describe quantities with both magnitude and direction. This makes them different from scalars, which have magnitude only. Common examples include displacement, velocity, and force, where direction matters. A vector is shown as a directed line segment from an initial point to a terminal point. It may also be written with bold letters or with an arrow above the symbol. Two vectors are equal if they have the same magnitude and direction, even when they start in different places. Vector...

the dot product

Dot Product From Direction and Angle

The dot product measures how much one vector acts along another vector’s direction. Vectors are quantities with both size and direction. This calculation gives one number that shows how aligned two vectors are. One way to find the dot product is to combine the horizontal parts and the vertical parts of the two vectors. The matching parts are multiplied, and the results are added together. This form shows why the value depends on both direction and strength. The dot product becomes larger when...

key terms and concepts

Learn how vectors use magnitude, direction, components, unit vectors, and the parallelogram rule for addition, subtraction, and scalar multiplication.

Vector Components and Direction | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Vector Components and Direction

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