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 \( k \).
- Set the first components equal: \( 4 = 4 \) (This is always true and doesn't provide information about \( h \) or \( k \)).
- Set the second components equal: \( 2-k = 2h-2 \).
- Set the third components equal: \( 7 = 7 \) (This is always true and doesn't provide information about \( h \) or \( k \)).
- 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:
- \(4 = 4\) (siempre cierto, no proporciona información sobre \(h\) o \(k\)).
- \(2-k = 2h-2\)
- \(7 = 7\) (siempre cierto, no proporciona información sobre \(h\) o \(k\)).
- \(2-h = -k\)
Resolvemos el sistema de ecuaciones:
- \(2-k = 2h-2\)
- \(2-h = -k\)
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:
\[
\boxed{h = 2}
\]
\[
\boxed{k = 0}
\]
Answered
