Problemas Resueltos - Razonamiento matemático
Problema 81
Si \( \vec{x} = (-1, \, 3) \), \( \vec{y} = (2, -1) \) y \( 3 \vec{x} + 2 \vec{y} - \vec{w} = (8, \, 5) \), entonces, necesariamente \( \vec{w} \) es igual a:
-
\( (-1, \, 2) \)
-
\( (-7, \, 2) \)
-
\( ( -9, \, 6 ) \)
-
\( ( 7, -2 ) \)
-
\( (-1, \, 6) \)
Intenta resolverlo antes de ver la respuesta...
-
\( (-7, \, 2) \)
En efecto:
\[
\begin{aligned}
3 \vec{x} + 2 \vec{y} – \vec{w} = (8, \, 5)
&\Rightarrow
-\vec{w} = (8, \, 5) -3 \vec{x} – 2 \vec{y}
\\[2em]
&\Rightarrow
\vec{w} = -(8, \, 5) + 3\vec{x} + 2 \vec{y}
\\[2em]
&\hspace{2em}=
-( 8, \, 5 ) + 3 ( -1, \, 3 ) + 2( 2, -1 )
\\[2em]
&\hspace{2em}=
(-8, -5) + (-3, \, 9) + (4, -2)
\\[2em]
&\hspace{2em}=
( -8 -3 + 4, -5 + 9 – 2 )
\\[2em]
&\hspace{2em}=
\boldsymbol{ (-7, \, 2) }
\end{aligned}
\]
\[
\begin{aligned}
&3 \vec{x} + 2 \vec{y} – \vec{w} = (8, \, 5)
\\[2em]
&\hspace{1em}\Rightarrow
-\vec{w} = (8, \, 5) -3 \vec{x} – 2 \vec{y}
\\[2em]
&\hspace{1em} \Rightarrow
\vec{w} = -(8, \, 5) + 3\vec{x} + 2 \vec{y}
\\[2em]
&\hspace{3em}=
-( 8, \, 5 ) + 3 ( -1, \, 3 ) + 2( 2, -1 )
\\[2em]
&\hspace{3em}=
(-8, -5) + (-3, \, 9) + (4, -2)
\\[2em]
&\hspace{3em}=
( -8 -3 + 4, -5 + 9 – 2 )
\\[2em]
&\hspace{3em}=
\boldsymbol{ (-7, \, 2) }
\end{aligned}
\]