Questions: para que valores de h y k el vector (4,2-k, 7,2-h) ES IGUAL Al VEctor (4,2 h-2,7,-K) ?

To determine the values of \( h \) and \( k \) for which the vectors \((4, 2-k, 7, 2-h)\) and \((4, 2h-2, 7, -k)\) are equal, we need to set the corresponding components of the vectors equal to each other. This will give us a system of equations that we can solve for \( h \) and \( k \).

1. Set the first components equal: \( 4 = 4 \) (This is always true and doesn't provide information about \( h \) or \( k \)).
2. Set the second components equal: \( 2-k = 2h-2 \).
3. Set the third components equal: \( 7 = 7 \) (This is always true and doesn't provide information about \( h \) or \( k \)).
4. Set the fourth components equal: \( 2-h = -k \).

Solve the resulting system of equations for \( h \) and \( k \).

Para que los vectores \((4, 2-k, 7, 2-h)\) y \((4, 2h-2, 7, -k)\) sean iguales, sus componentes correspondientes deben ser iguales. Esto nos da las siguientes ecuaciones:

1. \(4 = 4\) (siempre cierto, no proporciona información sobre \(h\) o \(k\)).
2. \(2-k = 2h-2\)
3. \(7 = 7\) (siempre cierto, no proporciona información sobre \(h\) o \(k\)).
4. \(2-h = -k\)

Resolvemos el sistema de ecuaciones:

1. \(2-k = 2h-2\)
2. \(2-h = -k\)

De la ecuación \(2-k = 2h-2\), simplificamos para obtener:

\[ k = 2h - 4 \]

De la ecuación \(2-h = -k\), simplificamos para obtener:

\[ k = h - 2 \]

Igualamos las dos expresiones para \(k\):

\[ 2h - 4 = h - 2 \]

Resolviendo para \(h\):

\[ 2h - h = -2 + 4 \\ h = 2 \]

Sustituimos \(h = 2\) en \(k = h - 2\):

\[ k = 2 - 2 = 0 \]

Los valores de \(h\) y \(k\) que hacen que los vectores sean iguales son:

\[ \boxed{h = 2} \] \[ \boxed{k = 0} \]