Chen Hilton
04/07/2024 · High School
Tomando los vectores a y \( b \) y el escalar \( k \) definidos como: \( a=(3,1,6), b=(-1,2,3) \) y \( k=2 \) determine el grado de verdad de la expresion (1) \[ k a \times b=a \times k b \]
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Para determinar el grado de verdad de la expresión \( k a \times b = a \times k b \), primero calculamos los productos vectoriales de los lados de la igualdad.
Damos los vectores y el escalar:
\[ a = (3, 1, 6) \]
\[ b = (-1, 2, 3) \]
\[ k = 2 \]
El producto vectorial (o cruce) de dos vectores \( \mathbf{u} \) y \( \mathbf{v} \) se define como:
\[ \mathbf{u} \times \mathbf{v} = \left( u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1 \right) \]
Primero calculamos \( k a \times b \):
\[ k a = 2 \cdot (3, 1, 6) = (6, 2, 12) \]
\[ (6, 2, 12) \times (-1, 2, 3) = \left( 2 \cdot 3 - 12 \cdot 2, 12 \cdot (-1) - 6 \cdot 3, 6 \cdot 2 - 2 \cdot (-1) \right) \]
\[ = (6 - 24, -12 - 18, 12 + 2) \]
\[ = (-18, -30, 14) \]
Luego calculamos \( a \times k b \):
\[ k b = 2 \cdot (-1, 2, 3) = (-2, 4, 6) \]
\[ a \times (k b) = (3, 1, 6) \times (-2, 4, 6) \]
\[ = \left( 1 \cdot 6 - 6 \cdot 4, 6 \cdot (-2) - 3 \cdot 6, 3 \cdot 4 - 1 \cdot (-2) \right) \]
\[ = (6 - 24, -12 - 18, 12 + 2) \]
\[ = (-18, -30, 14) \]
Ambos lados de la expresión resultan ser iguales:
\[ k a \times b = (-18, -30, 14) \]
\[ a \times k b = (-18, -30, 14) \]
Por lo tanto, la expresión \( k a \times b = a \times k b \) es verdadera.
Quick Answer
La expresión \( k a \times b = a \times k b \) es verdadera.
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