0.2
Producto vectorial o cruzado
La multiplicación de vectores produce un producto vectorial, con la magnitud igual al producto de los vectores individuales multiplicado por el seno d…
El producto cruzado o vectorial de dos vectores es el producto de sus magnitudes y el seno del ángulo. Se dirige perpendicularmente al plano de los vectores.
Geométricamente, es el área del paralelogramo atravesada por los vectores.
Usando la regla de la mano derecha, si el dedo índice y el dedo medio de la mano derecha señalan el primer y el segundo vector, respectivamente, el dedo pulgar señala su producto cruzado.
El producto cruzado del segundo vector con el primero tiene la dirección opuesta. Por lo tanto, el orden importa para los productos cruzados.
La medida de la rotación de una puerta resulta de un par que es el producto cruzado de la distancia radial entre la bisagra y el punto de aplicación de la fuerza con la fuerza aplicada.
La puerta no gira si una fuerza actúa a lo largo o en dirección opuesta a la distancia radial.
Para el vector unitario de un eje, su producto cruzado consigo mismo es cero. Pero con otro, está el tercero en orden cíclico.
Las componentes del producto cruzado de los vectores A y B son AyBz menos AzBy, AzBx menos AxBz y AxBy menos AyBx.
View the full transcript and gain access to JoVE Core videos
Q1: What is the cross product and how does it differ from the dot product?
The cross product is a vector operation that produces a new vector perpendicular to two input vectors, unlike the dot product which yields a scalar. The cross product's magnitude equals the area of the parallelogram formed by the two vectors, and its direction follows the right-hand rule. This operation is fundamental in physics for calculating torque, angular momentum, and magnetic force.
Q2: How do you calculate the magnitude and direction of a cross product?
The magnitude of a cross product is found using the formula |A × B| = |A||B|sin(θ), where θ is the angle between vectors. Direction is determined by the right-hand rule: point your fingers along the first vector, curl them toward the second, and your thumb indicates the result's direction. This ensures the cross product always points perpendicular to both input vectors.
Q3: What are the key properties of cross products in vector mathematics?
Cross products are anti-commutative, meaning A × B = −(B × A). They are distributive over addition and associative with scalar multiplication. The cross product of parallel vectors is zero, while perpendicular vectors yield maximum magnitude. These properties make cross products essential tools in three-dimensional vectors analysis and engineering applications.
Q4: When is the cross product used in mechanical engineering applications?
The cross product calculates torque (force applied at a distance), angular momentum, and magnetic forces on charged particles. Engineers use it to determine rotational effects in machinery, analyze forces in three-dimensional systems, and solve problems involving perpendicular force components. Its ability to produce directional vectors makes it invaluable for complex mechanical design and analysis.
Q5: How does the right-hand rule determine cross product direction?
Point your right hand's fingers along the first vector, curl them toward the second vector through the smaller angle. Your thumb then points in the direction of the cross product. This consistent rule ensures predictable directional results and prevents ambiguity when calculating cross products in three-dimensional space.
Q6: Why is the cross product only defined for three-dimensional vectors?
The cross product requires three dimensions to produce a vector perpendicular to two input vectors. In two dimensions, there is no unique perpendicular direction; in higher dimensions, multiple perpendicular directions exist. Three dimensions provide the unique geometric relationship necessary for the cross product's physical interpretation and engineering applications.