Exercise 11

If \( \vec{v} = (-2, \, 3) \)   and   \( \vec{w} = (3, \, -1) \), then   \( -2 \vec{v} - (-3) \vec{w} \)   equals to:

  1. \( (2, \, 3) \)

  2. \( (-5, -9) \)

  3. \( (5, \, 9) \)

  4. \( (13, -6) \)

  5. \( (13, -9) \)

Try to solve it before checking the answer.

  1. \( (13, -9) \)

Indeed:

\[
\begin{aligned}
&-2 \vec{v} – (-3) \vec{w}
\\[.5em]
&\hspace{1em}=
-2 ( -2, \, 3 ) – (-3) (3, -1)
\\[.5em]
&\hspace{1em}=
-2 (-2, \, 3) + 3(3, -1)
\\[.5em]
&\hspace{1em}=
( -2 (-2), -2 (3) ) + (3 (3), 3(-1))
\\[.5em]
&\hspace{1em}= (4, -6) + (9, -3)
\\[.5em]
&\hspace{1em}=
(4 + 9 , -6 -3)
\\[.5em]
&\hspace{1em}= \boldsymbol{ ( 13, -9 ) }
\end{aligned}
\]
\[
\begin{aligned}
-2 \vec{v} – (-3) \vec{w}
&=
-2 ( -2, \, 3 ) – (-3) (3, -1)
\\[.5em]
&=
-2 (-2, \, 3) + 3(3, -1)
\\[.5em]
&=
( -2 (-2), -2 (3) ) + (3 (3), 3(-1))
\\[.5em]
&= (4, -6) + (9, -3)
\\[.5em]
&=
(4 + 9 , -6 -3)
\\[.5em]
&= \boldsymbol{ ( 13, -9 ) }
\end{aligned}
\]

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