2.9
Producto Vectorial (Producto Cruz)
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.
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Q1: What is the cross product and how is it calculated geometrically?
The cross product of two vectors equals the product of their magnitudes and the sine of the angle between them, directed perpendicular to their plane. Geometrically, it represents the area of the parallelogram spanned by the two vectors. The magnitude depends on both vector sizes and their angular relationship, while direction follows the right-hand rule.
Q2: How does the right-hand rule determine cross product direction?
Point your right hand's index and middle fingers along the first and second vectors respectively. Your thumb then points in the cross product direction. If you reverse the vector order, the thumb points opposite, showing that cross products are anti-commutative. This rule resolves the ambiguity of two perpendicular directions.
Q3: Why does the order of vectors matter in cross product calculations?
The cross product is anti-commutative: reversing vector order reverses the result's direction. The cross product of vector B with vector A points opposite to the cross product of A with B. This directional reversal means A × B ≠ B × A, making order critical for correct physical interpretation.
Q4: What are the component formulas for calculating a cross product?
For vectors A and B, the cross product components are: (AyBz − AzBy), (AzBx − AxBz), and (AxBy − AyBx). These formulas use vector components in the Cartesian coordinate system to compute each component of the resulting cross product vector systematically.
Q5: When is the cross product of two vectors equal to zero?
The cross product equals zero when vectors are parallel or antiparallel, including when a vector is crossed with itself. Since the sine of 0° and 180° is zero, any vector pair aligned in the same or opposite direction produces zero cross product, regardless of magnitude.
Q6: How does torque relate to the cross product in physics?
Torque is the cross product of the radial distance vector from a hinge with the applied force vector. A door rotates only when force acts perpendicular to the radial distance; force along or opposite the radial direction produces zero torque. This demonstrates how cross product direction determines rotational effect.
Q7: What is the cross product of unit vectors in cyclic order?
Unit vectors of perpendicular axes follow a cyclic pattern: each unit vector crossed with itself yields zero, but crossed with another unit vector yields the third one in cyclic order. This property simplifies calculations and reflects the fundamental orthogonal structure of three-dimensional space.